In what ways would it be different from solving a human murder case? And what circumstances would make killing robots a serious offense, even if they don't have rights and are still referred to as manufactured property?
You could put it under Vandalism or Destruction of Property laws. If these robots were somehow written as being sentient that might raise moral issues, but not legal ones based on them being property. You could make it more serious by having the crime treated as an "attack on the owner by proxy," something like a Hate Crime tied to the human owner.
Or if you want to really twist meanings for a story, the destruction of "manufactured property" could fall under the legal umbrella of "conversion," which is "a civil wrong (tort) in which one converts another's property to his/her own use, which is a fancy way of saying 'steals.'" By destroying the robot you have stolen its use from the owner.
So in this way it's a theft, and assuming that robots are expensive, it would be equivalent to Grand Theft Robo or something similar, in a very roundabout but potentially interesting way.
I can't think of anything crime more serious because it's pre-determined that it's only property.
A complication in solving the case might be that robots literally do all look the same, again based on your world. Can the detective prove that this robot wasn't some other junk robot ready for the shredder?
As for other differences: robots may have recordings being captured even after "death," or most certainly right up until death shuts one down. Whether robot recordings are given as much weight in courts as human eyewitness testimony is up to the world and story, recordings are faked all the time... What if it were other robots that recorded, they are property and might not be able to testify.