When is it OK to use stranded wires in fixed installation?
All wire sizes are available in stranded, however not all cabling styles are made with stranded wire (i.e. NM/Romex). Any sort of flexible cordage is only stranded. 

For single-wire 10-12-14 gauge THHN/THWN as is widely used in commercial work, they sell both kinds right next to each other - watch out what you buy.  I use exclusively stranded.  

Larger wiring sizes are not available in solid, because they are impossible to bend.  

You can use stranded wire anytime you can get it, and your terminations are listed for stranded wire. For instance a backstab (push-in) connector might be listed for ”14 AWG solid" in which case they mean "not stranded".