I think the rules cover this pretty well already.
Initially, the chance for detection would be Passive Perception (at -5 due to disadvantage). This covers all of those minute possibilities that they are just noticed. Once there is a reason for somebody to actively be searching, I would rule that they make a Perception check with disadvantage.
Note that even without the boots of elvenkind, invisibility automatically grants advantage on your attack (because you can't be seen), and then is dispelled. If your opponent is attempting to attack first, it's at disadvantage (because they can't see you), and might miss altogether if they don't know where you are.
The miss chance goes away if they know where you are (succeed on their Perception check). Once they know where you are (presumably even if you have a ring or cloak that can't be dispelled), you'd have to successfully Hide (with advantage, and their Perception check is probably at disadvantage) in order to be someplace else to maintain that miss chance.
The fact that all of this is irrelevant with a 1st level spell (Faerie Fire) can't be overlooked. Throwing a shovelful of ashes from the fireplace onto the floor, smashing a few bottles, or similar approaches to reveal footsteps, or even throwing ashes or flour onto the invisible creature (DMs judgement as to whether it becomes invisible) are also things to keep in mind.
Another thing that DMs often forget is that your allies can't determine where you are either. So if they are planning on rushing into battle (or firing missiles) then there could be potential consequences as well. At the very least I'd probably rule that a critical miss might in fact have struck the silent and invisible ally. It depends a lot on the action that's going on.
As was also mentioned before, other sense still work, and I think that, at least in their home, most high level villians have far too few pets (dogs, etc.) than they should. Once a dog has latched onto the invisible creature, then Perception checks are irrelevent. Attacks against them may still be at disadvantage, but they wouldn't have a miss chance then either.
But by far my favorite defense is Faerie Fire. I point it out to every new spell caster because it not only reveals the invisible creature but everybody gets advantage on attacks against them. An almost ridiculously powerful 1st level spell.
Ilbranteloth