Damaged fishingrods when attempting to enchant are given the basic layout for enchantent but when one is chosen nothing occurs, i tried it with a undamaged one and it was able to be enchanted.
UPDATE: Fishingrods when put in a enchant table in the most recent snapshot (14w31a) are given options for enchants now such as unbreaking but ALSO include unlabeled enchants which cannot be applied.
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Not fixed.
Put a damaged sword on the enchanting table -> Can be enchanted
Fishing rod -> Cannot be enchanted.
Enchantments are not all adapted to the fishing rod. Therefore, when you select one randomly, it may be that the rod doesn't enchant.
Cannot confirm for 14w31a, seems fixed

Still in 14w31a
Fishing rods in an enchantment table with the cursor over the top most enchant gives no enchant forehand (I.E. Unbreaking ; Luck of the Sea). Other one which is applicable is ubreaking.

Is this still a concern in the current Minecraft version? If so, please update the affected versions in order to best aid Mojang ensuring bugs are still valid in the latest releases/pre-releases. If this has been done, we can reopen the issue.
Keep in mind that the "Resolved"-Status on this ticket just means "Answered", and that we are waiting for further information on whether this issue still exists or not. We will reopen it as soon as the requested information has been delivered.
Ticket resolved as incomplete, because no answer in a reasonable amount of time (1+ year), if it still happens, please update the ticket.
Fixed in 14w11b. Damaged fishing rods can be enchanted.