Damn this ae giving me a headache.  I have removed them from 3 of my kits and only messed up 3 drill bits..  I have been drilling them out and then hitting them with a pin punch. I have broken a coupe ad they keep getting stuck.  I have drilled them out with 3/16 and 11/64.  the punchs often get stuck.  This has been a major pain in y ass.   Are there any tricks that i am mising?
Thanks
Jimmy

I use a drill press and clamp the trunion securely. Then I start with a smaller bit, like 1/8 and lots of cutting oil. Drill halfway, a little at a time, flip the trunion, drill through. Make sure you don't just blast the hole; drill a little then bring the bit up to clear the shavings and add some more oil. Then switch to a slighly larger bit and repeat. And then a larger bit, etc...  Make sure your bits are sharp. Obviously don't work up to a bit that's too big for the rivet hole.

Eventually the rivet will spin out and leave a nice clean hole. No punching needed.

Lots of cutting oil and allow the bit to cool. No oil and blasting a hole all the way through will bust a bit for sure.

Edited because my spelling-fu is weak tonight.

I keep breaking punches and found a cheaper way. Grind rivits flush and then drill one side almost all the way through and then use big nails (10 penny ?) put in hole and hit with a 3lb hammer.

It does ruin the nail though.  

yes drill halfway thru then flip it over that way u wont be to far off from center AND u dont have the middle of the bit wanting to break
 i never had any trouble til the last one i did...reason i had trouble was i got in a hurry!

Second that last post, I got into a hurry on my first build and busted a bit next to my hand. Took me two weeks to dig out all the little pieces. Two lessons: dont hurry; use common sense

5/32" cobalt drill bit and a drill press.  Drill 90% of the way through.  Tap it out.  Works every time for me.  Just make sure you drill level and right through the center.  Its a good idea to grind the rivet flat and then mark the center with a center punch so your drill bit does not drift when you start drilling.