I use a curved perma-lok type needle bought from Tandy about 45 years ago. Lost it a couple times and luckily found it again. I decided recently to get a couple extra needles. The new ones are way too thick. My old one works for 3/32 and 1/8 lace with no problems. I'm going to take a file to the new ones and see how much I can remove from the tip. The tips look like an elephant stomped on them!
I have found good perma-lok needles are far better than the spring leaf types with a couple barbs. They just don't hold on for tight pulls.
If you break the lace off flush with the end of a perma-lok, heat it up with a match or lighter and burn the lace out of it. Tap the ash out, clean off any soot, and go back to work. It has only happened to me once, so I am a little more careful about how tight the pull is. It it's too tight, I enlarge the slit a little and make sure the layers are properly glued and lined up.
Happy lacing!