She wants to try the new ice cream.
She wanna try the new ice cream.
Are both equally common in everyday speech? What's an example where "wanna" would sound weird or out of place? (Still, in informal speech.)
In certain dialects of AmE, notably urban black dialects, "wanna" is correct (in that dialect) 3rd person singular. From the article on African-American Vernacular English on Wikipedia:
Present-tense verbs are uninflected for number/person: there is no -s ending in the present-tense third-person singular. Example: She write poetry ("She writes poetry").
He wants to try the new ice cream.
He want to try the new ice cream
So it would not be surprising in the contraction you mention that the urban dialect would yield
He wanna try the new ice cream.
"wanna" is a common pronunciation of "want to". The "nn" is not usually pronounced like "n", but is rather a nasal flap -- a very short sound. "wanna" is not used when there was originally a word or phrase between the "want" and the "to" ("Louise is who Joyce will want [Louise] to/*wanna eat ice cream").
For me, "wanna" sounds normal in (1) and weird in (2):
(1) Who do you wanna visit?
(2) Who do you wanna visit Fred?
In linguistics, we would normally put a * in front of the sentence in (2) to indicate that it is weird, and that we think it's weird because it is syntactically ill-formed. Note that (2) is not mean to be a question addressed to Fred, e.g. (3):
(3) Who do you wanna visit, Fred?
(3) is of course not at all weird, but it's not what is meant in (2). Instead, it's the same sentence as (1) with the addressee's name tagged on the end. These examples come from this website, which offers a clear explanation for the unacceptability of (2).
I would guess that (2) is a weird use of "wanna" for all native speakers of English, regardless of dialect, but that is an empirical claim, and the only way to know for sure would be to test it out with an experiment.