Although the OPed question does not seem to be concerned with overlapping occurrences, they can also be handled with m{pattern}g. If the sub-string may contain regex metacharacters, it is also wise to meta-quote.

>perl -wMstrict -le
"my $substring = 'xxx';
 for my $string (@ARGV) {
   my $non_overlap =()= $string =~ m{      \Q$substring\E   }xmsg;
   my $overlap     =()= $string =~ m{ (?= (\Q$substring\E)) }xmsg;
   print qq{$non_overlap non-overlapping $substring in $string};
   print qq{$overlap overlapping     $substring in $string};
   }
"  x  xx  xxx  xxxx  xxxxx  xxxxxx
0 non-overlapping xxx in x
0 overlapping     xxx in x
0 non-overlapping xxx in xx
0 overlapping     xxx in xx
1 non-overlapping xxx in xxx
1 overlapping     xxx in xxx
1 non-overlapping xxx in xxxx
2 overlapping     xxx in xxxx
1 non-overlapping xxx in xxxxx
3 overlapping     xxx in xxxxx
2 non-overlapping xxx in xxxxxx
4 overlapping     xxx in xxxxxx
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Update: The capturing group in  m{ (?= (\Q$substring\E)) }xmsg is needed only if there is a concern with what matched rather than only with how many matched as in the OPed question.