You could have caught this yourself if you would have used strictand warnings:

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

my @needles   = ("1-2 Steps", "5-7 Steps", "8-10 Steps", "catch");
my @haystacks = ("move 1-2 steps", "move 8-10 steps to hoist", "catch 
+the ball");

for my $needle (@needles) {
    my $found;

    for my $haystack (@haystacks) {
        if ($haystack =~ /\i\Q$needle\E\i/) {
            print "<br>";
            print "Found [$needle] in [$haystack]\n";
            print "</br>";
            $found++;
        }
    }

    if (!$found) {
    print "<br>";
    print "Couldn't find [$needle] anywhere!\n";
    print "<br>";
    }
}
[download]

Results in Perl yelling at you for using \i:

S:\Steve\Dev\PerlMonks\P-2017-05-31@0243-RegEx-Case-Insensitive>perl r
+egex0a.pl
Unrecognized escape \i passed through in regex; marked by <-- HERE in 
+m/\i <-- HERE 1\-2\ Steps\i/ at regex0a.pl line 12.
Unrecognized escape \i passed through in regex; marked by <-- HERE in 
+m/\i1\-2\ Steps\i <-- HERE / at regex0a.pl line 12.
<br>Couldn't find [1-2 Steps] anywhere!
Unrecognized escape \i passed through in regex; marked by <-- HERE in 
+m/\i <-- HERE 5\-7\ Steps\i/ at regex0a.pl line 12.
Unrecognized escape \i passed through in regex; marked by <-- HERE in 
+m/\i5\-7\ Steps\i <-- HERE / at regex0a.pl line 12.
<br><br>Couldn't find [5-7 Steps] anywhere!
Unrecognized escape \i passed through in regex; marked by <-- HERE in 
+m/\i <-- HERE 8\-10\ Steps\i/ at regex0a.pl line 12.
Unrecognized escape \i passed through in regex; marked by <-- HERE in 
+m/\i8\-10\ Steps\i <-- HERE / at regex0a.pl line 12.
<br><br>Couldn't find [8-10 Steps] anywhere!
Unrecognized escape \i passed through in regex; marked by <-- HERE in 
+m/\i <-- HERE catch\i/ at regex0a.pl line 12.
Unrecognized escape \i passed through in regex; marked by <-- HERE in 
+m/\icatch\i <-- HERE / at regex0a.pl line 12.
<br><br>Couldn't find [catch] anywhere!
<br>
S:\Steve\Dev\PerlMonks\P-2017-05-31@0243-RegEx-Case-Insensitive>
[download]

Two ways to fix this.

Two key changes:

my $needle_regex = quotemeta $needle;
if ($haystack =~ /$needle_regex/i) {
[download]

New code:

my @needles   = ("1-2 Steps", "5-7 Steps", "8-10 Steps", "catch");
my @haystacks = ("move 1-2 steps", "move 8-10 steps to hoist", "catch 
+the ball");

for my $needle (@needles) {
    my $found;
    my $needle_regex = quotemeta $needle;

    for my $haystack (@haystacks) {
        if ($haystack =~ /$needle_regex/i) {
            print "<br>";
            print "Found [$needle] in [$haystack]\n";
            print "</br>";
            $found++;
        }
    }

    if (!$found) {
        print "<br>";
        print "Couldn't find [$needle] anywhere!\n";
        print "<br>";
    }
}
[download]

Or, in keeping with the approach you originally took, only one change needed:

if ($haystack =~ /\Q$needle\E/i) {
[download]

Resulting in:

my @needles   = ("1-2 Steps", "5-7 Steps", "8-10 Steps", "catch");
my @haystacks = ("move 1-2 steps", "move 8-10 steps to hoist", "catch 
+the ball");

for my $needle (@needles) {
    my $found;

    for my $haystack (@haystacks) {
        if ($haystack =~ /\Q$needle\E/i) {
            print "<br>";
            print "Found [$needle] in [$haystack]\n";
            print "</br>";
            $found++;
        }
    }

    if (!$found) {
        print "<br>";
        print "Couldn't find [$needle] anywhere!\n";
        print "<br>";
    }
}
[download]