What is the safety carrying load of a 20 amperes circuit breaker?

A 20 amp circuit breaker can carry 20 amps. However a 20 amp circuit breaker requires 12 gauge wire or larger. A 12 gauge wire can carry 20 amps. However, there is an additional restriction, "a dedicated branch circuit rating shall be not less than 125% of the non-continious load." What that effectively means is that a dedicated branch circuit can carry 80% of its rating.

May I ask why the question?
Is this for a specific application or just a general question?

A breaker primarily protects in an overcurrent situation. People think a 20 Amp breaker will automatically shut off at 20 Amps. That is not the case, I have measured more than 20 Amps flowing through a 20 Amp breaker. The bigger issue is the wiring insulation on a continuous feed at 20 Amps, it can thermally deteriorate and burn off, possibly causing a short, at least in a grounded metal box, at which point the breaker will(should) trip. I have replaced quite a bit burned wire in a plastic box and raceway where the breaker never tripped, the wire melted and broke the circuit. See HKStroud comment regarding circuit loading.

If you exceed 80% of the breaker capacity it starts to heat up. Breakers can safely carry their rated capacity, but that should be avoided. Overheating and premature failure is what you can expect at near capacity.