Lee is correct, in California, there is a statute that you can press charges against someone for looking through your trash and taking stuff. The police won’t do anything about it unless you actually press charges or if they make a horrible mess in the process.
As for the police and FBI looking through trash for clues and evidence of crimes, they’re allowed to do that as part of an investigation, and are allowed to do it without a search warrant if they are searching through one of those big dumpsters often found in alleys and such. They are not allowed to search through private trash cans that are on private property without a warrant or probable cause.
If they’re patient enough to wait for the garbage collectors to come pick up your trash, as soon as it gets tossed in the truck, it’s considered to then be the collectors’ property. So, the police can ask them, “May we look in your truck?” immediately after they’ve picked up garbage from the house in question, and anything they find in the bags is admissible in a court of law. Well, assuming the collecters’ say it’s okay, but since most of them are run by the city, or at least funded by the city, they probably won’t say no.
Here’s the latest word on going police going through the garbage in Washington State: We can’t touch it while it is still in your trash cans. Once it’s dumped in the truck, there is no expectation of privacy, and we can dig through it all we want. It’s common to ask the trash guy to compact what’s in his truck before getting to the target house, so the stuff you want isn’t mixed in with the neighbor’s garbage.
My understanding is Fourth Amendment case law (covering the Feds and many states other than Washington) allows search of the cans once they are placed on the curb.