I use YLI nylon monofilament, which is very fine. It comes on a cone, which I keep in a small mason jar behind my machine with a thread cover on it (stolen from my serger stash). There are some tricks to using a fine monofilament like this. If you decide to go this route, check previous threads (no pun intended!) for additional info, or I can post again.
I should add that I have tried Invisafil and a polyester monofilament from Superior Threads for invisible machine applique, and the problem is that they are just not as invisible as the YLI.