There are a few assumptions here, and I'll try to note them (and will update if comments point out ones I miss). Primarily, I'm going to use the perspective of a developed, relatively wealthy nation. I also assume that magic is, for the most part, legal (like personal gun ownership in many countries).
You mention that the world is otherwise "much like our modern world", so a robust government system that can test and license magic users would exist. This licensing would identify ability level and skill, much like different driver's licenses for different vehicle types. In order to gain "magical" employment, you must be licensed. Since it's a rare skill, many people born with the skill will get licensed (and take courses/testing) to open economic opportunity. Enforcement would include a requirement to produce your license in various situations, including during investigations where the police may show up at your door; failure to comply may speak for itself depending on the society. This leads us to...
If magic emits radiation, and magical skills differ, I assume there is a magical fingerprint. This may be unique to an individual, or at least not universal. This would allow associating a usage to an individual. If licensing is in place, the gov would have that fingerprint on file. If fingerprints are general, the license would identify the magic users' various skills, and allow searching by fingerprint type.
Magical societies/rights groups
Much like there is MENSA and such, I'm sure there'd be magical societies. With magic being rare, I'm sure these societies would have a vested interest in keeping magic in a positive public light, meaning they'd work with police to suss out a criminal. This can also manifest as magical rights groups that advocate for the rights and place of magic users within society.
Assuming magic can be detected, I'll make a very large assumption it can be suppressed (ie. anti-magic effects). While these would be expensive, rich governments/corps/individuals would implement the solution for highly sensitive areas. This could be remote or local (that is, area devices or worn devices). Magical prisons would also be outfitted with these devices, and located in areas with low risk of escape due to location (try swimming from Easter Island, or walking from the South Pole to civilization). Others have already mentioned SWAT (MWAT?) so I won't retread that.
There would definitely be laws surrounding the use of magic in general society concerning contracts, public safety, legal proceedings, intelligence, etc. As others have mentioned, these would be followed by most people. Minimum sentences, or entirely unique laws, would exist for things like "Assault with a magical act". However, just like in the real world, 3-letter gov agencies would have special carve outs exempting them from the laws, and those agencies and govs would have the best stuff (as others have mentioned). As such, magic users (criminal or otherwise) would have their own concerns for safety, with fear keeping them on the straight and narrow.
Most of this deals with law-abiding citizens, and that is because most laws assume lawfulness. You cannot legislate people to not be criminals; you can only define what is a criminal. In that case, look to the law enforcement trends around weapons, cults, mafias, extremist groups, and such for further inspiration of how to deal with powerful criminals.