Match regex and assign results in single line of code

Why do you want it to be shorter? Does is really matter?

$variable = $1 if $variable =~ /(find something).*/;

If you are worried about the variable name or doing this repeatedly, wrap the thing in a subroutine and forget about it:

 some_sub( $variable, qr/pattern/ );
 sub some_sub { $_[0] = $1 if eval { $_[0] =~ m/$_[1]/ }; $1 };

However you implement it, the point of the subroutine is to make it reuseable so you give a particular set of lines a short name that stands in their place.

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Several other answers mention a destructive substitution:

( my $new = $variable ) =~ s/pattern/replacement/;

I tend to keep the original data around, and Perl v5.14 has an /r flag that leaves the original alone and returns a new string with the replacement (instead of the count of replacements):

my $match = $variable =~ s/pattern/replacement/r;

my($variable) = "some string" =~ /(e\s*str)/;

This works because

If the /g option is not used, m// in list context returns a list consisting of the subexpressions matched by the parentheses in the pattern, i.e., ($1, $2, $3 …).

and because my($variable) = ... (note the parentheses around the scalar) supplies list context to the match.

If the pattern fails to match, $variable gets the undefined value.

Well, you could say

my $variable;
($variable) = ($variable = "find something soon") =~ /(find something).*/;

or

(my $variable = "find something soon") =~ s/^.*?(find something).*/$1/;

You can do substitution as:

$a = 'stackoverflow';
$a =~ s/(\w+)overflow/$1/;

$a is now "stack"