How the duce would they ever find the perpetrators?
There are various stages: someone would have to report a criminal or suspicious act, perceived through some traces that remained after the fact; then someone would have to make the decisions to initiate an investigation; then that investigation would have to collect meaningful data; then that data would have to be matched to other data by someone (or something) who (or that) cared (or was programmed to treat it as relevant); then once the investigation "solved" the crime, at least to the stage of creating a list of meaningful suspects, then the suspects would have to be apprehended.
Some of the issues are techinical, some are jurisdictional, and some are matters of productivity and scale.
Begining at the beginning, what traces are detectable would depend on the crime: fingerprints, recording of emmissions, DNA, voice stress analysis, micro-changes in air density....the mind shudders. A homicide leaves a body. Some crimes are not effectively detectable in retrospect: if I take a legally owned firearm, for instance, and make a temporary, illegal modification that leaves no physical trace, without being contemporaneously received, and I reverse that modification before it is detected, then I would have commited a crime that leaves no trace.
Some crimes would leave traces, but the crime is not serious enough to warrant forensic collection.
Let's say, for instance, that your hypothetical is something like, "I shoot belter in the back with my laser rifle, while I am on an EVA over his claim, he dies, and there are no witnesses. How would [whomever may care] catch me?" Well, you might give yourself away, if it weren't random violence. Did you know him? Did you stand to benefit from his death? Did you take anything from the scene? As to forensic terminal ballistics on lasers, I'm a bit spotty, however just as the details current forensic terminal ballistics might have been impossible for one at TL 2 to guess, likewise there may be some technology able to measure the laser to its wound/damage/residue. What sort of remote sensing was present in the belt? On the claim? Video recording? Nearby small craft? A lot of variables. Usually, most investiations are social, relying on people who run their mouths, as most do. As Mickey Rourke's character said in a B Movie, "In any serious crime there are about 100 things that can go wrong; if you can think of 50 of them you're a genious."