Answer:
I was personally taught 11 exceptions to obtaining a search warrant. I've chosen the four that I think are the most common.
(1) Consent – A warrant is not needed if the person gives the officer their permission to search their things without one. If an officer asks "may I search your car" and the driver says "yes", a warrant isn't needed.
(2) Plain View – If the objects are in "plain view", a warrant isn't required to conduct a search. An officer doesn't need a warrant to observe that which can clearly be seen in a place where there is no reasonable expectation of privacy.
(3) Evanescent Evidence – If the officer has reason to believe that a person will destroy the evidence in the time it will take to get a warrant, places may be searched without one.
(4) Hot Pursuit – Any attempt a person makes to physically escape before the officer is done with them is grounds for arrest without a warrant, and anything the officer comes across, sees, or walks by in their pursuit of them, is automatically searchable without a warrant.