This post brought to you by a special request from the Marquis over at the East Coast Guide to Living. The Marquis asks...
"Just exactly what was the role of the local constabulary in apprehending criminals throughout the Middle Ages?"
The answer to this, like any question which addresses the medieval period as a single continuous block, is very complicated and we'll have to take it apart by period. Since I know a bit of Late Classical stuff we can start there, with the Roman version of the constabulary and move forward through time to examine when and were guards got their infamous positions making sure there was peace in any given city or town (hint: very late or very early, but not in between).
The Romans. I have no idea what the Hellenistic world, the Persians, the Babylonians, or the Egyptians did in terms of guards and constables. I do, however, know how the Romans played that game. They had both the vigiles and the Urban Cohorts to act as a sort of police force. These were paramilitary organizations which primarily kept riots and things from taking control of the Republic. The vigiles kept a night watch as well, to prevent or stop robberies and catch runaway slaves. These are as close to the fantasy trope of "guards" as we're going to get for a long time.
The Early Middle Ages. Say farewell to guards. They don't exist. The closest thing there might are maybe some reeves or ealdormen in England, perhaps a local bailiff in Frankia. When something goes wrong, it is the duty of citizens to apprehend the culprit. The reeve or ealdorman (depending on your time period in England) will put offenders up in a croft or manse to be held for judgement if they've committed a crime that the lord has to come and adjudicate or, (ENGLAND ONLY) if they've committed a high crime which requires waiting for the royal assizes to travel through the area and assemble a trial.
Towards the end of this period you have a lot of people pulling "guard duty" or something like it in castles. It's part of the services they owe their lord, including paying for their own armor and poles or truncheons. These guys are overseen by one or more knights. They aren't trained, they're just your average local schmucks except they have a leather jack on now. These guys aren't in cities, just at fortifications. Cities are still utterly guardless.
The High Middle Ages. None here either, though the organization of a night watch occurs in england towards the end of the 13th century. This is a commoner-composed force just like the castle guards. People get chosen to serve for a month and then they have to be a sort of makshift vigiles squad. Knights or serjants of some kind usually accompany them.
The Late Middle Ages. Retinues of princes start developing more police-like duties. However, the true police force (one that investigates, stops, and prevents crimes) doesn't exist for another several centuries. So stop running your watch like a modern police department, you dope! Or keep doing it, that's fine. I have very few "police" like figures at all in the 10th Age, myself.