I can't say for every small town, but I grew up in and still live near a town that size.
The department has 14 officers - 1 chief, 1 assistant chief, 3 shift sergeants, 1 criminal investigator (detective), 1 narcotics officer, 7 patrolmen, and 2 clerks. The criminal investigator handles all major crimes but I believe all sworn officers (except for the newest hires) have training in investigative procedure and would handle at least the initial investigation to any crime they responded to.
There also is a county-wide SWAT team composed of officers from all the local departments who respond to situations requiring their special training.
Evidence gathered at a crime scene is usually passed on to the State Police for processing and evaluation. And the State Police would be called in for a crime - say a multiple murder - beyond the capabilities of the local department.
Some towns in our area have merged their departments into one larger department protecting all the towns involved. Some have disbanded their departments and rely on the State Police for law enforcement.
The nice thing about fiction is you can make your small town police department however you want it to be. A chief and a bunch of officers who do everything. A chief and several detectives (each with a specialty) and a slew of patrol officers.
I'm writing about a town about the same size and have a chief, assistant chief, detective, and 12 patrol officers in the department. But I had it and the surrounding communities pool their resources to establish a county crime lab.
Hope this helps.
Dave