What are two types of searches that do not require a warrant?

Actually, there are more than two. Although some may be debateable as to whether they are even searches at all (stop & frisk and inventory), the Supreme Court has consistently ruled them as constitutionally legal.

Consent - self explanatory

Stop & Frisk - reasonable suspicion - safety

Vehicle - probable cause to believe the vehicle contains contraband or evidence

Exigent Circumstances - ie, imminent harm to another/incidental to arrest, evidence being destroyed, person screaming from inside a dwelling.

Inventory (search) - impounded vehicle

1

Searches of a person incident to custodial arrest do not require a warrant. Also, police generally do not need a warrant to search a vehicle if: (1) the person is not under arrest and is within reaching distance of the interior of the vehicle; or (2) the police reasonably believe that there is evidence of the crime the person was arrested for contained in the vehicle. See Arizona v. Gant (2009), USSC. Police may also conduct an inventory search of an impounded vehicle if the department has standard guidelines for the search and the officer conducts the inventory search in good faith adherence to those guidelines.

Also, the police are authorized to conduct any type of search if the person to be searched voluntarily consents to the search.

a Consensual search - that you gave the cops permission to do so

a search based on probable cause ( Detection of Odor of illegal substance)

a cop in pursuit of a suspect who runs into a home, or into a car, can search the car (or home) should the suspect drop any evidence

in a public place a cop doesn't need a search warrant

consentual and probable cause?

immenant harm?

there have to be more than two.