Not medieval, but I borrowed from my dad an interesting old book, the Middlesex County Records, Old Series, Volume I, 1549-1603, published by the Greater London Council, 1972, reprinted from the original edition of 1886. It's just a list of crimes from the old Middlesex County archives from the reigns of Edward VI to James I. I suppose these crimes were recorded as part of the court proceedings at the time. Full details are given for each crime.
The first thing that strikes you is the extraordinary violence and brutality of the crimes at the time. I suppose everyone in those days carried a knife, and I think if you were a gentleman you were allowed to carry a sword. Fights, duels and quarrels resulting in deaths seem to be extremely common.
The next most common type of crime recorded is theft and robbery, and every sort of item is stolen, from horses to clothing to jewellery.
Rapes and sex crimes are also recorded, with the details always left untranslated from the Latin. The book contains a number of cases of the rapes of children.
I know this isn't an answer to the question, since the period covered by the book is after the Medieval period, but I can't think the types of crime recorded in this book would've been so very much different from those committed 100 or so years before.
If you like I can give you some examples of indictments from this book verbatim.
There is a wealth of information when it comes to laws and court cases from medieval times. Several of the earliest of these are law books and codeces from the Frysian and Carolingian era, which give an amazing insight into the sort of crimes that would occur in the towns and communities of that period.
Like in most periods of history, the most common crimes seem to have been theft, assault and murder. The Lex Frisionum, recorded in the late 8th century, for example has a large variety of specific fines for specifics types of theft or injuries, as well as different fines for when the crime was committed by and done to a nobleman, a freeman/freewoman, serf or slave. (Note that the clergy is not mentioned as a category since they were only liable to church law.) It sometimes goes into seemingly absurd detail (at least to modern eyes):
On Manslaughter If a noble man kills a noble man, he pays 80 solidi; of this compensation, two parts belong to the heirs of the victim, and a third to his next of kin.
And if he denies, to have killed him, he swears it with 11 men of the same rank.
If a noble man kills a freeman, he pays 53 solidi and 1 denarius [= tremisse]; and if he denies, he swears with 7 oath-helpers.
If a noble man kills a serf, he pays 27 solidi minus 1 denarius to his lord, and [of this] to the victim's kin 9 solidi minus one third denarius; and if he denies, he excuses himself with three oath-helpers.
But also:
He who hits a harp player, who can play harp with a circle on the hand, pays a one quarter higher fine, than for another man of the same rank. Likewise for a gold smith.
If someone kills the slave of another man, he pays him according to the value at which his lord estimates him.
Likewise, horses and cattle, sheep, goats, pigs and any other livestock that belongs to men, including the dog, will be paid according to the owner's assessment.
[The killing of] a hawk dog or a small beagle, which is called barmbraccum [lap-beagle] is fined with 4 solidi.
But if it can kill the wolf, with 3 solidi.
In Frysian law, at least, nearly all crimes had wergeld as punishment, which was basically a fine paid to the wronged party (or its lord in the case of serfs and slave) and occasionally the king. However, some crimes had harsher punishments:
If anyone breaks into a shrine and steals sacred items from there, he shall be taken to the sea, and on the sand, which will be covered by the flood, his ears will be cleft, and he will be castrated and sacrified to the god, whose temple he dishonoured. (Note that this is most probably a leftover from pre-Christian law in Frysia.)
So, to answer your first question: crime was basically what you'd expect, lots of theft and violence. But that's not really anything special compared to every other time period.
The most characteristic is the division of the people into categories to determine which law applies to you. If you lived in Frysia, it would be the Frysian Law. Other areas had their own codeces and their own subdivision based on class and position in society. If you were a clergyman, it was the ecclesiastical law, which was for a large part separate from secular law made by kings and noblemen.
What type of criminals there were is a tougher question to answer. Most court records are about land disputes and what we might call civil law suits today, instead of criminal law. Perhaps an expert on the subject could help you further with that however.
If you're looking for more things to read:
The Lex Frisionum, in English. (Sorry about the horrible 90's page lay-out.)
Albrecht Classen, Connie Scarborough, Crime and Punishment in the Middle Ages and Early Modern Age
Wendy Davies, Paul Fouracre, The Settlement of Disputes in Early Medieval Europe