The issue is that you are escaping the carat - you are trying to match ^https instead of https. See perlre and/or perlretut. The correct code might look like

#!/usr/bin/perl
use strict;
use warnings;

my $string = 'https://somewebsite.com/';
if ($string =~ /^https/) {
    print "https success\n";
}
if ($string =~ /\/$/) {
    print "trailing slash success\n";
}
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Note I have used an un-escaped carat as an anchor to require matches start at the beginning of the string - I assume this is where your confusion originated.

Depending on what you are trying to do, you may consider checking out some of the useful modules of CPAN, such as Regexp::Common::URI.