The ul did not disappear did it? It ended up showing a line of dots on the page, If I'm right.... That was supposed to happen.
Try and add this to the top just after the <?php
$row['service']='AM';
$row['award']='Winner';
This will output the with a list of years. The years would have the url you are looking for with theservice=AM instead of service=1.
There is a problem with your logic though.
1. You need someway to initialize $row['award'] to either Winner or Notables or Honorables. This got initiated when you wrapped the whole thing in the while loop
2. You cant have an alternating if statement with $row['award'] trying to see "if" it is Winner or Notables. You should try an elseif.
Without looking at your entire code and making some assumptions on what the problem might me I thing this is what you are looking for.
<ul>
<?php
$sql="SELECT * FROM <table> WHERE year>=2000 AND year<=2010 ORDER BY year DESC";
$result=mysql_query($row);
while($row=mysql_fetch_array($result)){
?>
<li>
<?php if($row['service']=='AM' && $row['award']=='Winner') { ?>
<a href="winner.php?service=<?php print $row['service'];?>&year=<?php echo $row['year'];?>"><?php echo $row['year'];?></a>
<?php } elseif($row['service']=='AM' && ($row['award']=='Notables' || $row['award']=='Honorable Mentions')) { ?>
| <a href="honorable.php?service=<?php echo $row['service'];?>&year=<?php echo $row['year'];?>&award=<?php echo $row['award'];?>">Honorable Mentions</a>
<?php } ?>
</li>
<?php } ?>
</ul>
Change the above to the name of the table from which you are pulling the data.
The above will work provided you have your table set up with the following columns in it -> year, service, award.
Please let me know how it goes.