There is a gamut of venders providing quite a range of background investigations or checks (BGI), so the ubiquitous I/O Psychologist response "it depends" is appropriate. It's been a while since I've been in this space. In my telecom days, the use of pre-hire BGIs were very limited. As I recall, there were laws and regulations preventing most forms of pre-hire BGIs. We mostly operated in a post-offer, pre-start model (i.e., job offers were contingent on passing the BGI).

A variety of vendors offer web-based automated BGIs (e.g., First Advantage, Checkr, Orange Tree, Hire Right) that will cover work history, criminal history, credit history/financial fraud, drug testing, etc. These services tend to faster and cheaper than conducting these checks in house.  I do not believe they offer the "door to door" interviews of neighbors, family, and friends that are conducted in many government organizations (e.g., the Federal Government). If the City conducts psychological evaluations, these will also not be offered by most BGI vendors.