How do you pronounce the word 'vagary'?

Solution 1:

It used to be pronounced vuh-GARE-i, but it's changed. As @Tim above stated:

(GenAm) IPA: formerly /ˌvəˈɡɛɹi/, now commonly /ˈveɪˌɡəɹi/

It's usually pronounced VAY -guh-ri, but there are still people who pronounce it the "other way", as can be seen here

Solution 2:

In Gilbert and Sullivan's Iolanthe (1882) the old pronunciation definitely held:

We are dainty little fairies
Ever singing, ever dancing;
We indulge in our vagaries
In a fashion most entrancing.

I have to admit that since 1882 lyricists seem generally to have fought shy of using vagaries in their rhyme-schemes, so as a way of dating pronunciation-change, it's a bit lacking.