To be clear: the form of the trope that I'm referring to describes a primitive professional police force that would enforce civil and criminal laws. This is usually in concert with jails and a judiciary system. Fairness and corruption vary between settings, but these are the basic considerations I'm thinking of.
I am not a scholar, but to my knowledge, these sorts of things didn't exist at all in the European Middle Ages. Villages had a general tradition called the "hue and cry," where someone in distress would make an undefined call and everyone nearby was duty-bound to respond and deal with the assailant. Cities might have something vaguely similar, but this was mostly to keep violent incidents from escalating and not to enforce common laws. Part of the issue was that there wasn't a defined concept of the rule of law, and most laws were civil (harm against another person) rather than criminal (harm against society in general).
Again, the previous paragraph is based on my limited research; I am very much an amateur.
What I'm wondering is whether the trope described in the first paragraph existed anywhere in the premodern world. The Roman Empire? Han China? The American Empires? There are lots of civilizations that I don't know about, and I wouldn't be surprised if I've missed something.
Thanks in advance!
To answer the basic question, as explained in this previous post by u/PartyMoses explains, there were often city guards of a sort. However, they were not systematically responsible for law enforcement like, for example, the guards in Skyrim.
But as long as there have been cities, there has been a need for people to walk around the city to check that everything is ok. If there was a fire, there needed to be a way to raise the alarm and organise a response. If there was a murder, there needed to be some way to at least try to find the killer. Someone had to close the gates at night. Someone had to make sure people weren't dumping their trash in the wrong place, like in the street. While pre-modern societies never had something as organised as a police force, they did have systematic ways of dealing with the challenges of maintaining law and order and responding to emergencies. They had to, or administration was not possible. The Romans had the vigiles, which you can read about here and here.
You might be interested in this previous answer by /u/alkibiades415 to a question regarding the Scythians in ancient Athens. They acted on behalf of 11 elected magistrates in Athens who were responsible for arrests, executions and public order. There isn't much evidence to go by, so it should be taken with a grain of salt, but it could be an example of a premodern, urban "police force".
The Flemish cities had a police-corps in the 14th and 15th century. In Ghent they were known as the White Kaproenen (Kaproen is a kind of hat which covered shoulders and the neck that could also be worn as a turban.) Brugge and Ieper had similar groups called the Red and the Blue.
These cities had a lot of "Freedoms" which were land, taxation, governing rights which had been granted by their feudal overlord and were kept in the Belfry. These cities were ruled by council's of rich merchant and guild families.
In Ghent the White Kaproenen arranged traffic during processions, arrested criminals and were bodyguard for the city councilmen and were sometimes deployed as a militia in times of need.
Since the Flemish cities were fiercely competitive against one-another and often had tax disputes with their overlord or conflicting interest this happened quite often. Ghent revolted about every 20 years between 1275 and 1580.
Rebelheden ende Vergaringhen. By Jan Dumolyn.