.* is almost never useful in a well constructed regex. Instead use a character set restricted to just the allowed characters, or a negated character set that lists characters that are not allowed. In your case the closing ' character is a good candidate for a negated character set:

use strict;
use warnings;

my @grabArray;

while (my $current_line = <DATA>) {
      next if $current_line !~ /Grab this\s+'([^']+)'/;

      push @grabArray, $1;
}

print join "\n", @grabArray;

__DATA__

  Grab this 'test1' (repeat/new)
many lines
  Grab this 'test2' (repeat/new)
many lines
  Grab this 'test3' (repeat/new)
[download]

Prints:

test1
test2
test3
[download]