It is possible he becomes bugged and you cannot speak with him about stealing Dibella's Statue. If this happens, go to the Temple of Dibella, steal the statue for him, and then bring it to him. When you do this, both that objective and the objective that you would have gotten from speaking to him will both be completed.
After giving the statue to Degaine, it may not appear in his inventory.
Sometimes when talking to Degaine after obtaining the statue, the quest will not complete and the statue will remain in the Dragonborn's inventory.
It is possible to ask for the quest again at the Silver-Blood Inn but Degaine will not have the speech option to start the quest, thus leaving the misc. quest stuck in your quest list.
If Degaine is killed after starting the quest, "Steal the Statue of Dibella", the quest will not fail. If the statue has already been stolen, it will remain stuck as a quest item can not be removed from the inventory.
PC The solution to this is to use the console command player.removeitem 0008F997 1