For red and black wire for thin test leads, I've been using very flexible 24 AWG wire from CPC/Farnell. It's very low cost (£0.99 for 5m lengths) and 'feels' great in use, too. It has 133 strands. However, it is PVC, so it has limitations like temperature range.
In terms of multi-color, I was looking for reasonable-priced flexible wire not for multimeter probes, but for logic probe type applications, and noticed an online store called
BNTECHGO, and placed an order for 26 AWG wire (they offer other sizes too). It seems to be silicone (doesn't melt when I apply a 330 deg C soldering iron to the insulation). In terms of stranding, the 26 AWG cable had 30 strands of 0.06 mm tinned wire.
They sell reels at what seems a fair price for low-volume use, but they also sell packs of 10 colors in 5 ft (1.5 m) lengths. Unfortunately, gray is missing from the 10-pack, so if the full color code is needed, a reel of gray would also need to be purchased.
On the negative side, the BNTECHGO wire is initially really stinky. After a month, the smell reduces, but it's not gone (but now, it only smells if you put your nose within a few inches of it). Maybe I'm being too picky, but it's still irritating enough that I'd prefer an odorless wire supplier, even if it costs slightly more, and I will check the normal distributors at some point.