Can they prevent you from staying in your home if they saw something that could be considered evidence?

It's perfectly legal for a cop who comes to your home for some other reason (to respond to a noise complaint or a domestic violence call, for example) to arrest you if they see evidence of criminal activity sitting out in plain view. No search warrant required in that situation. This is true even if the cop is standing at your front door and sees drug paraphernalia (for example) sitting out on your coffee table. Cop doesn't have to even enter your home if he can see contraband in plain view while he is standing at your open front door.