Pouvoir

Basically, you should translate “can” this way :

I can = Je peux
I can’t = Je (ne) peux pas
I could = je pouvais / j’ai pu
I couldn’t = je (ne) pouvais pas, je (n’)ai pas pu

I can’t find

Be careful with this sentence, “can” is not translated here, since it wouldn’t have the same meaning in french, so you should do this way :

I can’t find my keys = Je ne trouve pas mes clés

“Je ne puis” is for sure very formal.
We use commonly “j’ai pu” / “Je n’ai pas pu” (litterally “I was able” / “I was not able to do”) for the past time.

Actually, the use of the “Passé Simple” (Je peux, > Je puis) is formal.
We usually speak about past things with the “Passé composé” time. (Subject + Auxilliaire + participe passé)

If you are speaking about something that happened, but is done/finished today, you’d say : “Je pouvais” / “Je ne pouvais pas”.

Have I answered your question ?

Present tense use of pouvoir

The present tense is completely fine and idiomatic:

Bien sûr, je peux !

Je peux le faire.

Demain, je ne peux pas.

In the negative, it can even be made to sound more colloquial by omitting the “ne”:

Demain, je peux pas.

It’s omitting the “pas” that sounds stiff and formal:

Je ne peux le faire.

Or even worse but still correct:

Demain, je ne peux.

And of course:

Demain, je ne puis.

(Actually, I have difficulties imagining myself uttering “demain, je ne puis” without some pompous tone and hand gesture to go with it!)

The special case of “aucun”

But in a sentence like yours (“I can’t find it in any dictionary”), you cannot fit a “pas” so if you want to keep the present tense you have to say

Je ne peux le trouver dans aucun dictionnaire.

and, while it’s already much better than “je ne puis”, it stills sounds a bit like “je ne peux”, whereas

*Je ne peux pas le trouver dans aucun dictionnaire.

sounds childish or downright incorrect.

Getting rid of “pouvoir” entirely makes the sentence feel less formal, even without “pas”:

Je ne l’ai trouvé dans aucun dictionnaire. [My original suggestion]

Je ne le trouve dans aucun dictionnaire. [Suggestion from @undu]

Past tense and “je n’ai [pas] pu”

On its own, “je n’ai pas pu” also sounds fine but I suspect that it’s not because of the past tense but because of the “pas”. And of course, it’s just as difficult to use in a sentence with “aucun” as

*Je n’ai pas pu le trouver dans aucun dictionnaire

is also incorrect. While

Je n’ai pu le trouver dans aucun dictionnaire

suffers from the same undertones as the present tense version and still feels quite formal.

Conclusion

Depending on what you are trying to say, it might or might not be possible to construct a present tense sentence that fits the situation but there are certainly many uses of “peux” in informal settings.

“Je ne puis” is actually very antiquated and not used in modern French.

“Puis-je… ?” is sometimes used as an extremely formal, somewhat tongue in cheek, way of interjecting politely, or even jokingly, in a conversation:

“Puis-je vous rappeler?” for instance means “May I call you back?” but in a very formal manner.

“Est-ce que je peux vous rappeler?” is more common and usually pronounced “Ess keu chpeux vous rappler?” (Do not forget that most French people do not pronounce the “eu” sound so as to speak faster).