I am a full time GIS Programmer for St Louis County Police (Missouri).
The number one thing I work on are emergency management plans. Flood models, evacuation plans, security plans, critical infrastructure maps, sensitive populations, risk analysis maps (hazards overlayed with critical infrastructure and sensitive populations). My biggest project to date was laying out the new outdoor warning siren system (which saved us about $2.4M over the old system design). Response is a big part of my job when we have disasters, especially damage assessment and impact zone mapping, but disasters are rare for most areas (not for us).
Next after that is maintaining and most importantly automated error checking 911 related databases (MSAG, ALI, etc.) This includes maintaining mapping layers for the computer aided dispatch system. The spatial database and spatial database replications parts get pretty complex. Error checking has resulted in building some massive FME workbenches, especially for ESN/MSAG.
Next is crime reporting, namely incorporating geocoding with crime reporting so that I can prepare incident maps for the public and politicians. Heat maps actually are not that popular, but are used. I also collaborate with geospatial researchers to generate research datasets and do more detailed analysis (like getis ord gi* and scan statistics).
I also regularly assist with professional responsibility (internal affairs) to analysis AVL data to respond to citizen complaints.
I also regularly aid the planning and analysis units who are developing beat plans.
I occasionally do oblique and aerial photos for tactical unit planning. I do not do anything with high resolution surveillance, since that is a 4th amendment violation without a warrant and I cannot see anything obtained with a warrant. I am just starting to work on live aerial imagery from helicopters and mapping for those systems.
Police officers really like the idea of being able to map leads, e.g. parolees, suspect vehicles, etc. in the area of a crime. But, there are some significant privacy and 4th amendment issues involved in allowing officers in the field to do that kind of mapping. Even for someone trained in spatial analysis, there are a lot of assumptions that create analytic pitfalls where you will accidentally make someone a suspect who has no connection to the crime.
Mobile mapping is not a very big area. Quite simply, almost no one has phase ii/next gen 911 implemented yet. When they do, all you really do is use the same basemaps as dispatch with one more record.
This is different from in-car situational awareness. That is very popular, but almost always done with 3rd party applications (due to the dispatch interfaces involved). Still, there is a large amount of spatial data management behind situational awareness too.
So, if I were to pick topics in order, I would say emergency management, 911/computer aided dispatch, command situational awareness/common operating picture, beat plans, public crime mapping, internal crime mapping, in-car situational awareness, and unit support (e.g. tactical planning, professional responsibility, etc).
I would also highly recommend contacting Ryan Lanclos, Jennifer Shottke, John Beck, or Tom Patterson with ESRI for more ideas. I have worked with all of these people previously and highly recommend them.