I usually use the 1/8" or 3/16" because they are easier to find. I do sometimes need the 1/16" eyelets for metalwork or for smaller items like charms. Someone emailed me about finding 1/16" eyelets because she was having problems finding them. I usually buy them whenever I find them in odd places. My stable source is Volcano Arts. I just also found them here:
I just ordered some from them, so I don't know how they are in comparison to the ones from Volcano Arts.
I use a 1/16" hole punch first and punch a hole in whatever it is, put the eyelet face down on the table and put whatever it is on top - so the backside of the eyelet is facing up. I then use the eyelet setter from Volcano Arts and tap a few times lightly on the back of the eyelet. It works great. Here is the link:
I have also tried the spring eyelet setter from Fiskars. It works with the 1/16 and 1/8 (most of the time) but not with the 3/16 (at least for me). I use the crop-a-dile for the 1/8 and 3/16 sizes anyway.
If I am doing something harder than what a hole punch can punch through, I use my Dremel or a metal hole punch that I got awhile back from outsidethemargins.com.