Or, as an auxiliary answer for completeness’ sake that you should probably ignore: if the array is very large and the strings are long,

@array = @array[ do {
    my @lc = map lc, @array;
    sort { $lc[ $a ] cmp $lc[ $b ] } 0 .. $#array;
} ];
[download]

Makeshifts last the longest.

@array = qw(ABU gubu abu GUBU babu BABU);
@array = sort {
    if (lc($a) eq lc($b)) {
        $b cmp $a
    } else {
        lc($a) cmp lc($b)
    }} @array;
print "@array";

__OUTPUT__
abu ABU babu BABU gubu GUBU
[download]

Update: Oops, sorry, missed reliable ordering of same words with different case. This version does it, taking into account that you want lower case before upper case.

@array = sort { lc($a) cmp lc($b) || $b cmp $a } @array;
[download]

is fastest (about 6× faster)

Actually, you can’t make such a blanket statement. It might be 2 or 3× slower if the dataset is small, like the sample dataset in the OP’s post (I didn’t bother to check). But it can also easily be 100× faster if the dataset is humongous. Results will vary by dataset size, so you can’t give numbers.

Another aspect not to be forgotten is that it always takes twice the memory.

Makeshifts last the longest.

Many locales (but not the default C locale) sort this way, so if you use locale; then cmp or sort will do what you want. For example:

[am]king ~$ LC_COLLATE=hu_HU perl -we 'use locale; print join(" ", sor
+t(qw"ABU gubu abu GUBU babu Babu bubu")), "\n";'
abu ABU babu Babu bubu gubu GUBU
[am]king ~$ LC_COLLATE=hu_HU perl -we 'print join(" ", sort(qw"ABU gub
+u abu GUBU babu Babu bubu")),
"\n";'
ABU Babu GUBU abu babu bubu gubu
[download]