What Officers Can and Cannot Do
When law enforcement officers see what they reasonably suspect is criminal activity, it's their duty to stop and investigate. If their reasonable suspicion develops into probable cause to make an arrest, they can do so without an arrest warrant. Most criminal investigations (and, ultimately, arrests) involve searches performed without search warrants, which isn't necessarily a constitutional violation. But it may be a violation depending on the facts.
The following is a basic introduction to criminal procedure. It does not go into detail, but it may help shed some light on day-to-day criminal investigations, particularly of lower-level offenses like DUIs.
They CAN pull you over if they have a "reasonable suspicion" you've committed any offense, large or small. The most minor of traffic infractions in enough to justify stopping you.
They CANNOT pull you over simply based on a hunch. They need objective "articulable" facts to justify stopping you, such as a traffic (moving) violation.
They CAN temporarily detain you to investigate whether you’ve, in fact, committed the offense they suspect.
They CANNOT detain you longer than what’s reasonably necessary to complete their investigation. HOWEVER, if reasonable suspicion of another offense develops, then they can detain you longer to investigate the newly suspected offense. For example, a traffic stop for speeding requires only a short investigation. But, if during the stop, the officer smells alcohol, he develops reasonable suspicion of drunk driving. The officer now has a basis to further detain the driver for a DUI investigation.
SEARCH YOU IN PUBLIC?
They CAN search you if they have probable cause, or even just reasonable suspicion, that you're carrying a weapon or contraband. Similarly, they CAN search most of your vehicle with probable cause, or reasonable suspicion, that contraband is inside. To search your trunk, however, they need probable cause.
They CANNOT search you or your vehicle if there’s no reasonable suspicion to do so. However, there are a lot of exceptions. Also, they are always allowed to simply ask you for consent or permission to search. You are allowed to say no. Surprisingly, many people with contraband on them or in their vehicles agree to be searched, and end up getting arrested. That’s because law enforcement authorities are intimidating, and many folks struggle to say no.
They CAN arrest you if their investigation leads to "probable cause" you've committed a public offense in their presence. It's a higher standard than "reasonable suspicion", but not by much. In legal terms, it means it's more likely than not that you committed the offense. In other words, probable cause means what it sounds like: essentially, you probably did it, nothing more than that. Proving you probably committed an offense is a MUCH LOWER hurdle than proving, beyond a reasonable doubt, you did it. Proof beyond a reasonable doubt is the prosecutor's burden at trial.