I have used different types of "invisible" thread. Nylon monofilament is one type; polyester monofilament is another. With these two types of thread, brand name is *very* important.
For nylon monofilament (available in both clear and smoke), my favorite brand is YLI. This type of monofilament thread is fine, like baby hair, and about .003mm in diameter. SewArt is another good brand for nylon monofilament, but I was never able to find it and it seems to me they discontinued making it some time ago. I would not use any other brand. Many of the complaints about nylon monofilament come from years ago, when it more closely resembled fishing line than baby hair, and off-brands.
For polyester monofilament, a good brand is Superior. When I compared Superior's poly to my YLI nylon, I found that the polyester was more visible. Although polyester is supposed to be able to stand higher temperatures than nylon, when I used my Rowenta to iron over YLI nylon thread over and over with the iron on its highest setting, I could find no changes at all in the thread. It is said that polyester monofilament stands the test of time better in that it is not subject to degradation from light, does not become brittle with time, etc.; however, it is my suspicion that these old reported problems with nylon thread were probably related to older brands. So far I have never seen YLI nylon monofilament thread being charged with brittleness over time or any of the other problems it is supposed to have.
Invisafil isn't what I would call invisible, although it is very fine. Superior has a new thread called MicroQuilter that is 100wt polyester and I assume it is similar to Invisafil, but I haven't checked it out yet.
A lot depends on what you want to use the thread for, and what kind of trade-offs you want. I loved YLI nylon monofilament for years for my invisible machine applique, but I didn't particularly like the feel of the thread on the applique. I experimented with polyester monofilament, but it was only slightly softer than nylon and it was more visible. I also tried Invisafil, which provided softness; however, the color in the thread made it much more visible to my eyes.
I used nylon monofilament for only one small quilt and, although the quilting thread was indeed very invisible, again I didn't like the feel of the finished quilt. Nowadays, if I want "invisible" quilting, I go with a very fine polyester thread such as Bottom Line. It sinks into the quilt enough to be virtually invisible, and I am willing to trade some invisibility for softness and feel.