tl,dr: I recommend forgetting about setting $1 and using

my $match = /#(\S*)/ ? $1 : "";
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First, some assumptions since the specs aren't clear.

• You didn't define what "the string" is. I'm going to assume it's "a sequence of zero-or-more non-spaces" (since it can be foo but not foo bli bla). I'll also cover stricter patterns in my answer.
• Your example shows the "#" at the start of the searched string, but you didn't specify it had to be there. I'm going to assume it doesn't have to be at the start.

The following is a single match that produces the desired result:

• / ^ [^#]* \#? ( \S* ) /x;
  say $1;
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(Update: There's a much simpler approach using (?|pattern). See tybalt89's answer.)

If it doesn't have to be a single match, you could also use the following:

• /()/; /#(\S*)/;
  say $1;
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• /#(\S*)/ || /()/;
  say $1;
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That said, I think setting a variable other than $1 would be better. This allows us to use one of the following:

• my $match = /#(\S*)/ ? $1 : "";
  say $match;
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• my $match = /#\K\S*/ ? $& : "";
  say $match;
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And if it's acceptable to produce undef instead of an empty string when the pattern isn't found, you could even use the following:

• my ( $match ) = /#(\S*)/;
  say $match;
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These are far more readable and maintainable. Among other things, this makes them far less error-prone.

Also, these would be trivial to adapt them to match more restrictive patterns. For example,

• my $match = /#(\S+)/ ? $1 : "";
  say $match;
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• my $match = /#(\w+)/ ? $1 : "";
  say $match;
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Adapting the first version would be horrible.

• / ^ (?: (?! \#\S ). )* \#? ( \S* ) /x;   # /#(\S+)/
  say $1;
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• / ^ (?: (?! \#\w ). )* \#? ( \w* ) /x;   # /#(\w+)/
  say $1;
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