As far as I know you cannot make index() case insensitive.
I think the simpliest thing to do is to make you string case insensitive.
Ex.

my $lcstr = lc $mystr;
my $ind   = index($lcstr,"a");
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This way "a" matches the first instance of "A" or "a" in the original $mystr.
Hope that helps.

Originally posted as a Categorized Answer.