Only with your consent, with a warrant or if the police have reason to suspect that there is some dangerous instrumentality in your car, e.g., a bomb. A valid arrest is a prerequisite to a search incident to arrest in Washington. Even with a valid arrest, the police can only search unlocked areas of the car. To search a locked container, such as a glove box or trunk, the police must have a warrant.
In Arizona, the answer to this question recently changed. There is new caselaw that holds that a search of the vehicle prior to arrest for officer safety once the subject is outside of the vehicle and secured is not reasonable. If, however, the officer sees something in plain view that would cause a reasonable officer to be suspicious (a weapon or drugs as an example), then they can certainly investigate prior to arrest.
Once a person is arrested however, the vehicle can be searched during an inventory search prior to impound.
Sometimes people who get arrested confuse some issues relating to "arrest". For example, if you have already admitted facts sufficient to provide legal cause for your arrest, and the officer has captured that information on video tape or DVD, or via a back-up officer, the officer engaging in the search may not ANNOUNCE (verbally) that you are being arrested or put cuffs on your wrists. However, through conduct or actions by the officer, that conclusion is apparent (e.g., the officer tells you that you should have taken a cab home instead of driving after 12 beers, or you hear the officer call over his/her radio for a tow truck to come to this location). In such situations, an officer MAY start what he or she will call "an inventory search" of the vehicle that is about to be towed. The appellate courts of some states are very strict about requiring that formal words of arrest be given before such "searches incident to arrest," while others bend over backwards to NOT declare such searches of the trunk as being unconstitutional, saying that the cocaine in the trunk was subject to "eventual discovery", because you already admitted sufficient facts to justify an arrest.
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