In Canada:
We are allowed (and are supposed to) search every person who is arrested, for evidence and for anything that might harm the police or the arrestee or would allow them to escape (handcuff key). This kind of search is granted through common law.
The vehicle search would depend on the type of offence they are arrested for. If they are arrested for DUI you are allowed to search the area where they were sitting in the car and everywhere that was within their reach while in the car (which is almost anywhere but the trunk). If it is a drug trafficking charge the search can be extended to the trunk (case law supported). If the car was just nearby, you would need a warrant to search the car (for drugs).
-If the person is arrested in a house, once again you can search the immediate area they were in when they were arrested (like the couch cushions they were sitting on), any other searches must be done with a warrant. There are exceptions to this if there are exigent circumstances and if someone's safety is at risk (firearms in house duing domestic).
The law of search is a complicated one and there are several authorities under which we act. In Canada we have search authorities under
Common law
Liquor licence act
Highway traffic act
Criminal code
Fish and game act
Coroner's act
and others I can't think of right now. I would suggest you look up search authorities for your area and see what you get.
Case law has a huge impact on police searches. For example, just recently there was case law about what is called "hot pursuit". A drunk driver left an accident scene and went home. Normally there would not be much we could do if the person refused to come to the door when the police showed up (by the time they got a warrant the guy would be sober). Case law has now told us that if you know the person has arrived home (by talking to witnesses for example) within the last 15-20 minutes or so, you can call that hot pursuit and go into the house to search for the person, even if you personnally didn't see him/her run in.