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use warnings;
use strict;
...
while (my $line = <$infile>) {
  print "Matched '$1' in: $line" while $line =~ /($comb)/g;
}

>>> OP version
Matched 'regexes' in: I'm trying to search for several regexes in some
+ long files. To speed things up,
...
Matched 'regexes' in: This seems to speed things up, at least when the
+ regexes are just plain words.
Matched 'regexes' in: But: If I try input regexes which are anchored (
+"^word"), then suddenly it's
Matched 'that' in: (I could rewrite ^aaa|^bbb|^ccc as ^(aaa|bbb|ccc), 
+but it might be that only