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use strict;
use Data::Dumper;
...
{
    my @sorted = sort { uc($a) cmp uc($b) } @array;
}

Benchmark: timing 1000 iterations of Schwartzian, Simple...
Schwartzian: 16 wallclock secs (16.39 CPU) @ 61.01/s (n=1000)
...
              Rate Schwartzian      Simple
Schwartzian 61.0/s          --        -33%
Simple      91.3/s         50%          --