There are a few reasons your Neopet may not be showing up on the 1-Player Battledome list.
1) Your pet may not have any health points! If your pet has 0 health points you must heal them before they can enter the Battledome. You can heal your pet with a Healing Potion or other special foods.
2) Your pet may be too hungry! Only pets who are well fed will feel strong enough to battle.
3) Your pet may be sick. If your pet has contracted a disease, it will feel far too sick to battle. You can check if your pet has a disease at your Quick Reference page here: http://www.neopets.com/quickref.phtml
If your pet is sick you can find out how to help them by visiting the Hospital here: http://www.neopets.com/hospital.phtml
4) Your pet may be stuck in a battle! Visit the status page here: http://www.neopets.com/battledome/battledome.phtml?type=status to clear it! If this doesn't work you may also try un-equipping (and then re-equipping) an item from your pet on this page: http://www.neopets.com/battledome/battledome.phtml?type=equip
If you are still having trouble, please create a bug ticket but be sure to try all of the above fixes first as this will be much faster!
Keywords: plot, bd, battledome, stuck, not sick, not hungry, drop down, dropdown