Definitions are an ongoing problem, and folklorists - let alone people outside the realm of folklore studies - have a hard time agreeing on definitions. I'll respond with three comments (one for each of your terms), drawing on excerpts from my Introduction to Folklore, which I plan to e-publish in early September. On folklore:

The first place to start a discussion about folklore is with definitions. This is necessary, ironically, because folklorists and the people they study sometimes use various terms in dramatically different ways. The need for a workable definition is, at the outset, especially pivotal for the word “folklore.” Although people may intuitively know what this means, professional folklorists have been unable to agree on precise language to explain what the term describes. Definitions are different from individual to individual and from place to place, and the meaning of words can change over time. The controversy over definitions reveals a great deal about the history of the field and some of the difficulties it faces as a discipline.

Part of the problem with defining “folklore” comes down to whether the word refers to a subject matter or a method of study. The term was invented by William Thoms (1803-1885) in 1846 who proposed, as he put it, that “Popular Antiquities, or Popular Literature … would be most aptly described by a good Saxon compound, Folklore, – the Lore of the People….” (Dundes, 1965) Thoms was proposing that his peers recognize the collective traditions of the common folk of the United Kingdom, but by “common,” he did not mean the average person on the street. The concern of the day was that industrialization, or modernization as it is sometimes called, was transforming society, permanently changing rural, agricultural life. Traditions that had the appearance of being centuries if not millennia old were disappearing. This disturbed Thoms and many others, who hoped the term “folklore” would bring focus to a subject matter that warranted attention. Even before Thoms made his proposal to an English audience, two German scholars had virtually invented the academic approach to the study of popular oral traditions. Jacob and Wilhelm Grimm (1785-1863 and 1786-1859, respectively) had published their first volume of Kinder und Hausmärchen (Children’s and House Folktales) in 1812 and their Deutsche Sagen (Teutonic Legend) from 1816 to 1818. A series of books written independently by Jacob Grimm earned him the title of the father of folklore science. He published Deutsche Grammatik (Teutonic Grammar) between 1819 and 1837, and his Deutsche Mythologie (Teutonic Mythology) appeared in 1835. All of this happened before Thoms wrote his essay, and so it is clear that his invention of the term “folklore” did not occur without a context. Indeed, German and Scandinavian scientists, many the direct intellectual heirs of Jacob Grimm, would continue to shape the field of folklore studies, internationally, throughout the nineteenth century. Part of what inspired the Grimm Brothers was the need to identify a common Germanic folk culture. At the beginning of nineteenth century, Germany consisted of dozens of small, weak, independent states, and Napoleonic France seemed poised to obliterate Germany, militarily, culturally, and linguistically. Like Thoms, the Grimm brothers recognized that a changing world was devouring timeless traditions, but these young scholars also felt that the preservation of traditions could help their beloved German culture to survive the challenges it faced. Other countries would adopt their methods, but the inspiration varied. Nonetheless, many of the best folklore archives in Europe exist today where people sought to link folk ethnicity with a nationalist movement.

The term folklore originally referred to the traditions of the European peasant, but not necessarily to the study of that material. During the nineteenth century, much of Europe’s peasantry were still confined by a nearly medieval illiteracy, and yet antiquarians felt they preserved the crown jewels of a nation’s heritage. For enthusiasts such as Thoms, folklore meant simply the collective traditions of the people. As academic attention increasingly focused on this material, the term folklore came to mean the field of study for this subject.

Parallel to the emergence of folklore as an academic discipline was the development of ethnography. The simultaneous birth of the two academic fields caused many of the difficulties in defining folklore. Webster’s dictionary describes folklore as “the traditional beliefs, legends, customs, etc., of a people” and as “the study of such lore.” It defines ethnography as “a branch of anthropology dealing with the scientific description of individual cultures.” A problem occurs because there is little to distinguish “culture” from “traditional beliefs, legends, customs, etc.” In short, folklore and ethnography examine virtually the same thing, at least in the simplest terms. Folklorists are left to give their field distinction by insisting that either their methodology or subject matter is unique. There has been little agreement in the discipline regarding the best approach to define the exclusive turf of folklore. When Funk and Wagnalls published its Standard Dictionary of Folklore, Mythology and Legend in 1949, it could not find consensus on the meaning of the term folklore. It consequently offered twenty-one individually-authored definitions and left the reader to sort out the differences.

Initially, folklorists found distinction in the fact that they studied their own European traditions, while ethnographers examined other cultures, namely those that Europeans were encountering in the rest of the world. This is an awkward place to draw the line separating the two because it implies that the oral traditions of other people are somehow different from those of Europeans, and no credible scholar could promote such a concept.

Urban legends are the direct counterpart of testimonial or Migratory Legends, a type of legend (stories told to be believed as opposed to folktales, the oral novels to the folk, told as fiction to adults among the peasantry of Europe). Urban legends can deal with the supernatural, although in the modern world, this is usually restricted to ghosts, but they behave just like their earlier counterpart, the Migratory Legend.

Again, an excerpt from my Introduction to Folklore:

Like the peasant stories that preceded it, urban folklore offers life-saving information. It tells us when to be cautious about handling cereal boxes at the store because a clerk has Hantavirus and is spreading it by handling groceries. It tells us to stop that unseemly, greedy practice of checking coin returns in vending machines. It seems that evil, drug-using AIDS patients are leaving their infected needles in the returns with the hope of spreading disease throughout America – or at least among the greedy. And it warns us away from eating bananas because some have been found to be covered with flesh-eating bacteria from South and Central America.

Of course, none of these helpful hints have been found to be based on anything that is true. Between investigations by www.Snopes.com and the Centers for Disease Control and Prevention, one urban legend after the next has been proven to be false. But these and hundreds of other tales make for great stories, and people will always have a thirst for this sort of thing. There are also several newer forms of folklore that have emerged in modern North American and European culture. The following is an overview of various forms that is not inclusive, nor can it hope to be up-to-date since modern folklore consists of a dynamic well of creativity that continually spawns new stories that circulate rapidly.

and:

Utah folklorist Jan Harold Brunvand (1933- ) has published several books on urban legends, raising the subject to a level of awareness that it had not previously enjoyed. The media frequently publishes these stories as true, but now the Internet often reproduces them as well. At the same time, debunking websites seek to extinguish the fires of urban legends, resulting in an ebb and flow of folklore, creation and spread followed by debunking and retreat. Nevertheless, the urban legend is most at home as a truly oral form of folklore. People tell the stories to be believed just like their legendary predecessors.

Urban legends include a range of information on many subjects including live animals and microwaves, the dangers of beehive hairdos, the hazards of adolescent dating (particularly when homicidal maniacs escape from local mental institutions), and the possibility of becoming an unwilling organ donor while on vacation. This is clearly important information! With the internet, stories are often short-lived, in part because of debunkers in the media and on the Internet, but also because many urban legends describe new technologies or passing fashions. Thus, having poisonous spiders nesting in a sprayed-and-seldom-washed beehive hairdo would only be appropriate when that style was in vogue. Similarly, people born after 1975 are less likely to have heard the story of the old woman who thought it would be a good idea to dry her newly-washed poodle in the microwave: the story could only thrive when microwaves represented new technology, with all the angst that sort of thing can produce.

One of the most persistent forms of urban folklore involves legends about ghosts. Some, such as the well-known “Vanishing Hitchhiker,” are clearly urban legends of the sort that Brunvand collects, but in this case, the story appears to have pre-modern roots. The most common legends dealing with ghosts in general are what von Sydow would call memorates. These are personal recollections that families tell of their own histories about departed loved ones. There has been little collection of these stories because each example is so specific to the individual circumstance. Like the material related to traditional European folklore, it is possible to recognize that while the memorates may not be traditional, the beliefs behind them and the conventions of telling them are. This topic remains a fertile opportunity for research.

Because of various factors, Urban Legends can come and go quickly. As a genre, they are reminiscent of the pre-industrial Migratory Legends, but those stories lasted for decades if not centuries, while an Urban Legend can have a much briefer life span. Fortunately, the folk are extremely creative and new legends rise up to take the place of those that society has discarded. Keeping up with the constantly changing spectrum of stories makes folklore collection and analysis a challenge. Brunvand remains the best source on this genre of folklore.

On Myth, again, from my Introduction to Folklore:

Something also needs to be said here about myth. People use this term awkwardly. In a European context, myths tend to be the artificial constructs of ancient and Classical-era priests or literate people who sought to weave folk traditions into a comprehensive whole. The exercise often had political purposes, designed to provide diverse people with a single set of beliefs and stories. By reconciling similar traditions, the shared culture of these groups could be seen as more important than the differences, justifying the central rule of the king and his priests. Myth is also a way of organizing and reconciling folk traditions, which by their nature can be contradictory and highly localized. Myth tends, however, to make gods of supernatural beings, giving those powerful entities a status – for modern readers – similar to the Judeo-Christian-Islamic God, even when this comparison is not justified. Of course, it is also important to point out that myths were stories that were told – and then written down – and they were different from religion itself. Many myths were simply the shared cultural inheritance of a group of people.

In general, the word myth is best set aside when discussing more recent folk traditions, recognizing its proper status as a literary genre. Nonetheless, ancient documents recording myths can assist in understanding the history of various stories and beliefs. The authors of these texts were, after all, the first folklorists, and they were the only ones coming close to practicing the craft at the time.

Some folklorists carelessly use the term myth to denote those legends that deal with a fantastic, remote time. This primal era saw the creation of many familiar things such as day and night, fire, animals, people, mountains, and all other aspects of the present world. Folklorists properly refer to these stories as etiological legends explaining the origin of things. Sometimes, however, people interchange etiological legends with the word myth. The problem with this is that “myth” can imply something that is inherently wrong, linked to “primitive” superstitious beliefs. When the term “myth” is used for the folklore of existing cultures or for the traditions that were viable only a generation or more ago, it can take on an insulting, derogatory tone. It is best to reserve the word “myth” for ancient and Classical-era texts.

Obviously, /u/itsallfolklore/ will be able to deliver a more thorough answer.

But in rough, I think it’s accurate to say:

• Mythology is pretty close to the literary canon in the way it’s constituted – older stories written down in old texts (like the Eddas, or Metamorphoses), that get retold in different ways again and again and become sort of touchstones for later works. Mythology is explicitly religious, and I think I’ve seen at least one definition that’s something like: “Any religion not shared by the speaker.”

• Folklore is stories told by “the folk” – ordinary people. It can have mythological or religious elements, but isn’t necessarily about, for instance, the origins of natural phenomena or the doings of gods. I believe (and this is my past as a lit major showing through) folklore can be defined as stories that are told by one group of people and written down by another. The Brothers Grimm were recording an oral tradition, for instance. Where mythology enters the academy as religion, classics and literature, folklore seems to be more anthropological a category.

• Urban legend is a more sociological category. As far as I know, the term was coined by Jan Brunvand, who is a folklorist. He was fascinated by the way certain odd little narratives seemed to pass through many different media, always barely plausible and almost always sourced to “a friend of a friend” (or FOAF). In some ways, I think these stories reflect how folklore transformed at the dawn of the Information Age… newspaper columns and tabloid stories were a great source of transmission for these, and once email became commonplace, they spread like wildfire. They’re contemporary and not exactly fantastic (no brownies to come and do the chores if you leave out milk for them), but not exactly mundane either (who is honestly going to mistake a giant sewer rat for a Chihuahua?).

Most importantly, the categories overlap quite a bit, and the terms are really not used in the same way by different people consistently. Stories also drift between categories: Some myths seem to get retold as folklore, and some qualities of folklore seem to have survived as urban legend.