$userInput = <STDIN>;
chomp $userInput;
$userInput = uc $userInput;
[download]

can be written (more concisely) as

Also, I think there may be a better method to running each subroutine than using multiple conditional operators. Perhaps a dispatch table will do.

my %pbactions = (
    L    => \&list_phone_book,
    LIST => \&list_phone_book,
    A    => \&add_phone_book,
    ...
);

# Later on~

defined $pbactions{$userInput} ? $pbactions{$userInput}->()
                               : usage();
[download]

Untested lines of code, they're just to inspire.