For case-insensitive string search, use index (or rindex) in combination with fc. This example expands on the answer by Eugene Yarmash:
use feature qw( fc );
my $str = "Abc";
my $substr = "aB";
print "found" if index( fc $str, fc $substr ) != -1;
# Prints: found
print "found" if rindex( fc $str, fc $substr ) != -1;
# Prints: found
$str = "Abc";
$substr = "bA";
print "found" if index( fc $str, fc $substr ) != -1;
# Prints nothing
print "found" if rindex( fc $str, fc $substr ) != -1;
# Prints nothing
Both index and rindex return -1 if the substring is not found.
And fc returns a casefolded version of its string argument, and should be used here instead of the (more familiar) uc or lc. Remember to enable this function, for example with use feature qw( fc );.