I assume that you are asking for three circumstances in which a warrantless search would be legal because of an emergency. The Supreme Court has ruled that warrantless searches are legal in such cases because there is no time for an officer to secure a warrant. If officers were to take the time to get a warrant, some sort of harm would occur.
One example of this is when evidence might be destroyed. An emergency search can be made if the officer has probable cause to believe that a delay will allow a suspect to destroy evidence of a crime.
A second example relates to public safety. If officers believe that a search is necessary to prevent someone from being harmed, they may do so
Finally, there is an exception in cases where officers believe that they need to render aid. If officers have cause to believe that someone in a house needs immediate assistance, they may enter and search without a warrant.
In all of these cases, waiting to get a warrant could cause some sort of harm. Therefore, warrantless searches are legal.