These are authentic Cats costumes, made for the original West End and Broadway productions.
Carbucketty - I’m afraid these photos weren’t taken specifically for construction of the fluffies, but for the shoulder fastenings. Still, you can see the way the fabric is a small strip, tied in a knot, with the ends frayed considerably. Each knot is machine sewn onto the unitard in small, short lines of at most 1/2”.
Tantomile - This was a very old unitard, the texture of the fabric shows how the spandex threads had completely perished over time (or possibly a bleach wash) and the garment was completely shapeless. However, the fluffies are clear and in a notably different style to the London costumes.
The first stage is that the organza itself was painted much like the unitards, then cut into strips on the bias (diagonally), one edge is treated - I’m not sure if it’s been given a rolled hem, or just serged (overlocked). The treated edge is then sewn onto the unitard with the machine on the same stretch stitch as visible on the hems of the neckline etc.
This old costume also shows many repairs and additions - notably one fluffy is applied quite differently, a solid black fabric, sewn down the middle and ruffled on. I had to pin this back to get a clear photo, the fabric naturally folded and covered the stitching.
So, if you’re making a Cats costume, which version is “better”? The london version is simpler if you don’t have a sewing machine, as hand sewing the knots is as easy as machining, only more time consuming. Tantomile’s long lines of fluffies accent her design, but on other characters solid stripes of fur can look unnatural. The London knots can end up looking like tassels, or bows, if not given a brush through every now and then.
As with so many things in Cats costuming, both methods are equally authentic, they have different results, choose whichever you prefer for your costume!