Others have already pointed out that you want the /s option to make . match a newline so you can cross logical line boundaries with .*. You might also want the non-greedy .*?:
use v5.10;
my $html = <<'HTML';    
    <td class="fieldLabel" height="18">Activation Date:</td>
    <td class="dataEntry" height="18">
        10/27/2011      
    </td>
HTML
my $regex = qr|
    <td.*?>Activation \s+ Date:</td>
        \s*
    <td.*?class="dataEntry".*?>\s*
        (\S+)
    \s*</td>
    |xs;
    
if ( $html =~ $regex ) {
    say "matched: $1";
    }
else {
    say "mismatched!";
    }
(2020 Update) But I'd use Mojo::DOM and CSS Selectors to get the date. The particular selector may depend on the complete HTML source, but the idea is the same:
use v5.10;
use Mojo::DOM;
use Mojo::Util qw(trim);
my $html = <<'HTML';
    <td class="fieldLabel" height="18">Activation Date:</td>
    <td class="dataEntry" height="18">
        10/27/2011
    </td>
HTML
my $dom = Mojo::DOM->new( $html );
my $date = trim( $dom->at( 'td.dataEntry' )->all_text );
say "Date is $date";
If you have the complete table, it's easier to use something that knows how to parse tables. Let a module such as There's also HTML::TableParser handle all of the details:
use v5.10;
my $html = <<'HTML';
    <table>
    <tr>
    <td class="fieldLabel" height="18">Activation Date:</td>
    <td class="dataEntry" height="18">
        10/27/2011      
    </td>
    </tr>
    </table>
HTML
use HTML::TableParser;
  
sub row {
    my( $tbl_id, $line_no, $data, $udata ) = @_;
    return unless $data->[0] eq 'Activation Date';
    say "Date is $data->[1]";
    }
 
# create parser object
my $p = HTML::TableParser->new( 
    { id => 1, row => \&row, } 
    { Decode => 1, Trim => 1, Chomp => 1, } 
    );
$p->parse( $html );
There's also HTML::TableExtract:
use v5.10;
my $html = <<'HTML';
    <table>
    <tr>
    <td class="fieldLabel" height="18">Activation Date:</td>
    <td class="dataEntry" height="18">
        10/27/2011      
    </td>
    </tr>
    </table>
HTML
use HTML::TableExtract;
  
my $p = HTML::TableExtract->new;
$p->parse( $html );
my $table_tree = $p->first_table_found;
my $date = $table_tree->cell( 0, 1 );
$date =~ s/\A\s+|\s+\z//g;
say "Date is $date";