The spec you are interested in is called the "stranding" of the wire.
You will see it written on spec sheets as something like 7x36 or 19x40.
In the 7x36, it means there are 7 strands of 36 AWG wire. 19x40 means there are 19 strands of 40 AWG wire.
The jacket material is important for the flexibility as well. Silicone insulation is more flexible than a PVC.
If you want flexible, you want as many strands as possible. With 22 AWG, a lot of your options are limited to 7 strands. If you are willing to step to 20 AWG, there are more options available at 19x32.
Hope that helps.