You can find some information on Low Voltage wiring in NEC Article 720 "Circuits and Equipment Operating at Less Than 50 Volts.
Article 720 is relatively brief and basically states that conductors should not be smaller than 12 AWG. 12 AWG is some hefty wiring for the low voltage you are considering.
Article 725 covers Class 1,2 and 3 "Power Limited Circuits"
A Class 1 circuit is one that is under 1000 VA. The maximum VA you listed is 120 (12V * 10A). As long as you have your circuit properly fused or on a circuit breaker you can classify your circuit as a Class 1 under Article 725.
725.25 States that you can use 18 AWG and 16 AWG in an approved raceway or cable.
Typical house wiring of this type is called bell wire. This cable usually has a brown outer jacket and has solid conductors (one white and one red) It is also available with more than 2 conductors and is used for wiring of doorbells and thermostats.
If you anticipate regular circuit loadings of 10Amps I would consider using 16 AWG wire. Otherwise I would think 18 AWG would be fine.
Edit: I see that 725.23 " Class 1 Circuit Overcurrent Protection" states that circuit amperage for 18 AWG wire shall not exceed 7A. Circuit Amperage for 16 AWG wire should not exceed 10A