nanofrog, I'm familiar with the "known good" ones, but availability or price is a problem around here.
Where are you located?
If you go into your user profile and set your country, your nation's flag will display beneath your userID. Makes it so much easier to answer questions, especially regarding relevant product vendors.

Purchasing parts and consumables from distributors is the best way to avoid counterfeits IME. Good products may not seem inexpensive when you first see the price, but it's actually the best value in the end (actually works and keeps you from damaging/destroying PCB's; then there's the aggravation factor). It's possible to get genuine products on eBay, Aliexpress, and DX, but it's more likely you'll get a fake instead. Goot, Hakko, and Amtech get counterfeited a lot.
One trick you might find useful, is to toss these in as filler items on a component order so you don't get raped on shipping for a roll or two of solder wick, flux pen, soldering iron tips, ... types of consumables (don't weigh much). Mouser, Digikey, and so on will carry quality consumables. Being a bit flexible on brand helps, but compared to what you're using, the worst name brand will run rings around what you're currently using performance wise.
Regarding solder wick braiding itself, it's usually rather small gauge wire in a tight weave pattern. But that's only part of the story, as the flux is absolutely critical (must have enough, and it work properly). To give you an idea, Chemtronics is actually a bit sticky (I use rosin), and stores indefinitely IME (can't remember how old mine is, but it works just fine). Cheap stuff tends to have little to no flux. Longer length spools (or packs) is cheaper in the long run (25m, 50m, 100m spools/total length of spools when added).
Pic might help (not mine).
