"I give nothing to no-one" or "I do not give anything to anyone"

Solution 1:

The only existing answer is at best misleading.

"I do not give anything to anyone" is standard English

"I give nothing to no-one" is a double negative - not uncommon, but considered "non-standard".

"This humour doesn't hurt anyone" is standard English.

"This humour hurts nobody" is standard English.

Note that the only "non-standard" variant above is easily recast to full acceptability:

"I give nothing to anyone" is standard English.

There's no difference in meaning whether the negation is applied to the verb (give/do not give, hurts/doesn't hurt) or to the object (anything/nothing, anyone/nobody).