There isn't anything wrong with using #12 on a 15A circuit breaker or an extension of a branch circuit, provided, as Absolute mentioned, you have the right size wire nuts, terminals, etc. In some cases, you'd have to use #12 on a 15A circuit to overcome voltage drop.
You could theoretically have someone see the #12 and incorrectly assume it was a 20A circuit, but that's on the installer.
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While obviously, I would call it out as wrong in the opposite situation - a 20amp breaker with 14 gauge wire.
Interesting part is that the rated ampacity for #14 (using the standard 75 deg C table) is 20A**. The 15A circuit breaker restriction comes from a footnote in the ampacity tables in Article 310, and is based on overcurrent protection requirements.
** #14 would physically handle 20A, but it's still a no-no to use with a 20A breaker.
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All city codes allow 14-2 if it is a 15amp breaker.
Fuzzyfan asked if you could use #14 on a "power circuit." Nobody would argue that #14 works on a 15A, but some municipalities don't allow the use of #14 in residential applicatons. Others, in commercial applicatons. City of San Antonio, for example, requires #12 period. From the Chapter 10 amendments,
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Sec. 10-1403.1 General exceptions (all types of use and occupancy classifications).
1. No conductor smaller than 12 AWG copper or 8 AWG aluminum shall be used for individual branch circuits and feeder wiring.
The blanket all types of use/occupancy covers the R-1, R-2, etc. occupancies for single family houses and apartments/condos.
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As said the breaker amperage determines the wire minimum size.
For 15 and 20A branch circuits feeding multiple receptacles, yes. In other cases, not necessarily. Say for example you had a condensing unit with a minimum circuit ampacity of 62A. A #6 copper wire is good for 65A. Since there is no 62 or 65 amp circuit breaker (ignoring the European and adjustable trip stuff), it may be fed with a 70A circuit breaker.
Now let's really bake the OP's noodle and ask about running and attaching the wire and where the nailguards go.