In the last year or so I’ve been converted from calling { and } curly braces to calling them moustaches - I don’t often find I need to discriminate between the 2 types of moustache at it is mostly obvious from the context.
(I know this is old, but I have something to contribute.)
I’m surprised no one mentioned pronouncing { and } as executive brackets. I don’t call them that, but I thought it was pretty common (I call them braces).
One of my teachers at college would pronounce single quotes as “doink”. I believe he also pronounced double quotes as “double doink”.
$ # % {} * [] ~ & <>
Dollar sign
Pound
Percent
Left-brace, right-brace
Left-bracket, right-bracket
Tilde
Ampersand
Less-than, greater-than
But then, I’m old enough to remember…something…but I’ve forgotten it now… 
$ === ‘bling’
/ === ‘whack’
! === ‘bang’
I had a junior high math teacher that called them left { and right } Bob Hopes, because her drawing made them looked like him in profile.
Thank you , really nice explanation! Really appreciate it, gonna print it and use it in school.