I'm not sure, whether you are referring to a bodily waste (feces and urine) or thrash, so I'll cover both in this short entry. Let's start with the latter.

Regarding trash, the problem was far less important than nowadays, as people throughout Middle Ages, already nowhere as numerous as today (between 11th and 16th century the population of Europe went from 57 to 90 millions, today it is 740 millions) did generate almost a negligible fraction of modern waste. Due to labour-intensive process of fabrication evident in nearly every single production branch, all materials were rarely wasted. There were no disposable containers. All forms of scrap wood (one of the most common materials) could have been used for various purposes and ultimately serve as firewood. Food was scarce rather than wasted, so people tried to make use of everything and things that were not eaten along with not so fresh leftovers were commonly used to feed the ubiquitous pigs. The thing that would be readily associated with 'trash' were predominantly clay vessels: cheap, common and rather brittle.

Of course, not every bit of material was re-used or preserved and with thousands of people living in a city, the workshop waste, broken utensils, bones, heavily damaged textiles and other common waste slowly accumulated. They were usually disposed of in designated pits or simply thrown to latrines (thus latrine pits sometimes harbour very interesting artifacts). The aforementioned trash pits were usually located in the open spaces and close to the city walls. The regulation of 1469 from Prague stated that people could have been throwing their waste in the specific places closed to walls and the cleaning of the streets was to be conducted on rainy days if possible. The latter gives an impression that these refuse pits were located closed to a opening in a wall and the rainwater was facilitating removal of the waste outside the walls (the city itself in located on a small hill). If the pits were slowly filling, the trash was usually carted to a dumping site outside the city. Streets of the medieval were cleaned regularly and special cleaning was usually ordered after every market day and large festivity resulting in increased amount of waste and horse manure. This was usually conducted by hired workforce recruiting from the poorest citizens and sometimes even from local prisoners. Some forms of trash, such as clay, gravel, wood chips or hard kitchen waste (fruit stones, nut shells etc.) were often found in the foundations of building, used, along with the rubble, as a common filler or insulation. Sometimes, large amounts of such fillers were used to fill unused parts of vaults or cellars.

For farmers, no matter what their social level, waste matter was free fertilizer and would be collected and used as such. For those in castles and other fortifications, it was easier and less smelly, to make sure it was deposited outside the walls, hence long drops into the moat from a hole in the bedchamber for example. As for food waste, you have to recall that there were animals about; hunting dogs, pigs etc which would root around and eat anything they could find.

Generally speaking, waste matter was simply something which was part of life in general. In cities and towns, waste was discarded in the street (pigs roamed around there as well). Butchers were notorious for depositing animal parts in the street. There were sometimes edicts issued against such acts, but the fact they were issued more than once indicates the continuous nature of the problem.

As a broad rule, waste in the country was used, waste in the towns was a nuisance.