I was thinking as I watched Yami's debut in Toloveru the other day that there are a lot of anime series that have really appealing side characters, but due to their side character status they end up suffering from a lack of screentime compared to the main heroine. The most recent example I can think of is Hinagiku, who in spite of winning the Best Moe tournament still found herself consigned to just a half dozen or so episodes out of 52 where she got to take a leading role, but there are plenty of other examples. In no particular order (and I'm sure I'm forgetting plenty of other good examples) here's my list of the most under-utilized side characters in anime:
Katsura Hinagiku (Hayate no Gotoku): The most obvious choice--in spite of leading the manga popularity poll, in the anime she didn't end up getting a whole lot of chances to shine, even getting cut out of the class trip to the beach, create-o. Although at least she did get to show up in that episode in the eyecatches. If you asked otaku to vote on the 'side character you'd most like to see in their own spinoff series in a lead role', Hina-chan might well win. Personally, I'm still rooting for Hayate x Nagi since they have more history and I think more of a bond between them, but it's hard not to root for Hina too.
Yoriko (Da Capo): Yoriko could carry her own series with the holy triumvirate of nekomimi, maido, and doji-ko on her side, but aside from just being a moe character, as Stripey recently mentioned she has one of the more emotional scenes in anime towards the end of the series. If Da Capo ep 22 doesn't bring a tear to your eye, odds are it's because you're too preoccupied with the rising cost of the oil that pumps through your cold, black heart.
Kotori (Da Capo): Kotori's a character that in almost any other bishoujo series would almost certainly be the winner, by a mile--but in Da Capo in terms of screentime she's really in third or worse, well behind Nemu and Sakura. Even in DCSS, she starts out looking like she'll be the star, only to lose the screentime battle, badly, to Aisia of all people. Kotori truly has no luck in love or the limelight, but at least she can console herself with the fact she has legions of fans who wish there was a Da Capo: Kotori After Story animated.
Kyou and Tomoyo (Clannad): It figures that when Key finally breaks out of its tendency towards needy, emotionally stunted heroines, and produces Kyou and Tomoyo, they then end up completely getting the shaft on screentime in favor of characters like Fuuko. Fuuko's a good driver, I'm sure, but come on, who would you rather see a multi-episode arc feature, Fuuko or Kyou? Then Kyoani teases us with Tomoyo's Sunohara ass-kicking at the beginning, only to forget she exists for half the series. I'm holding out hope that one day, Mahou Shoujo Kyou will become a truly epic Clannad spinoff, but no, looks like we get more melodrama and death instead. Damn you, Key/Kyoani.
Machi (Nagasarete Airantou): Machi deserves her own series--with her ku-ku-ku she's like a female version of Kururu, always with some evil scheme underway. That and her use of Akane voodoo dolls is responsible for a lot of the laughs in the series, for me anyway. It's too bad with a cast that seemed to grow every other episode, Machi didn't get as much screentime as I'd have hoped.
Kiyone (Tenchi Muyo): The ultimate tsukkomi character, paired with omega baka Mihoshi for all eternity--sadly, it's been long enough there's no way she'll ever get a Space Detective Kiyone spinoff. What about Sasami? Sasami has so many spinoffs I don't think she counts as a side character anymore, between Pretty Sammy OVA and TV, the more recent Sasami Mahou Shoujo Club, she actually made the side character->main heroine transition other side characters can only dream of.
Multi (To Heart): Last but certainly not least, Multi is possibly the most popular side character in anime history--granted she gets a lot more screentime in Remember my Memories (which most people would rather forget), but in the original To Heart a two-episode appearance turned into fame for Hocchan, a million Multi references in other anime to this day, and figures, doujinshi, and other goods that probably add up to half of Japan's GNP. To this day when I pick a broom it's hard not to yell 'Doryaa!' and start running down the hallway.