There are numerous instances of GIS crime data available to the public, at least in the US, probably too many list and/or no one has done an entire audit of the Criminal Justice open data space.
The overwhelming majority of these datasets are at the local level, on the locality's gov site and/or the locality's criminal justice agency's site (if they aren't the same; most are). For example, Richmond, Virginia's police department shares its data on Richmond's government site. Some even have dedicated data portals specifically for Police/Criminal Justice data, though CJ is a stretch here: the vast majority of them are focused on where crimes occur, which are part of CJ, but there is much more to CJ then that. Furthermore, no crime dataset I've ever come across includes white collar crimes, so they are all inherently flawed. Not to mention that most of them don't publish data about internal affairs and/or complaints about them. I'll get off my soapbox, just needed to make those points about how flawed Criminal Justice Data is in America.
Here are a few from Virginia:
Crime Incident Information Center - Richmond Police
Newport News Police Open Data and NNPD Daily Reports
These datasets are what you seek however, they aren't 100% usable, as they need to be geocoded for GIS. That is almost the de facto standard I have seen around the Commonwealth and assume is the same for the rest of the nation. If you need help geocoding, there are a number of options out there, but in these cases, which tend to be smaller batches, I almost always default to using Google Spreadsheets and a Google Apps Script that does the geocoding for free, in the browser.
Here are some datasets that fit what you seek exactly:
Arlington County has always done a great job at sharing open data (though their latest approaches to FOIA make all of this suspect), and they geocode their locations for you, Arlington County Police Incident Log as well as provide necessary GIS data overlays one would need to dive more into their data, like Arlington County Police Districts Boundary Map.
Data.gov offers a crime tag you can search, which can shed some light into which localities are sharing and what they are sharing, however at this point I'm not sold that their is any main hub for GIS Criminal Justice data, hence the focus on localities here.
DOJ and FBI have their massive datasets and some do indeed have GIS data included, I've just always found them...lacking, to say the least. That said, looking at the FBI's UCR Crime Data Explorer, I just snagged Virginia's 2017 dataset, and it does have a locations CSV.
edit - You'll have to figure out the GIS portions, as I said these are incredibly lacking.
Here's FBI UCR GIS dataset
Lastly, if this is for the US, simply approaching your local Law Enforcement Agencies and/or local government is not necessarily a bad idea. Numerous GIS datasets have been released in this approach from what I've seen, some via FOIA, but most from simply engaging the officials and asking clearly/directly for what they want.
