my $file = /(1M01)(\_)(F)(\0+)(121)/;
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In the above line, you matching against the default "thing", i.e. $_ and assigning the result to tha variable $file - not what you want, probably. The parens in the regular expression mean captures which are to be found in $1, $2, $3, ... and so on.

Furthermore, the char '_' needs no escape, and \0 does match a NULL byte, not the number 0.

That corrected, the following prints those captures mentioned above, if there is a match:

my $file = '1M01_F00121.npt.gro';
if ( $file =~ /(1M01)(_)(F)(0+)(121)/ ) {
    print join( "\n",
        "\$1 = '$1'",
        "\$2 = '$2'",
        "\$3 = '$3'",
        "\$4 = '$4'",
        "\$5 = '$5'",
    ),"\n";
} else {
    print "'$file': no match\n";
}
__END__
$1 = '1M01'
$2 = '_'
$3 = 'F'
$4 = '00'
$5 = '121'
[download]

Read perlop and perlre.