I'd like to fly somewhere for a weekend bicycle ride. Is it possible to attach my bike helmet to a small backpack (one of those used for a 2L water pouch) and call it a carry-on? (The bike itself is already at the destination.)
From an airport security standpoint I think the helmet should cause no problems in your carry-on. The TSA prohibited item search tool does not mention bicycle helmets. It does however mention all other types of helmets including motorcycle helmets, which are considerably tougher than bicycle helmets and can be used as an offensive weapon. Interestingly enough, TSA says that motorbike helmets are allowed in carry-on luggage:
The same is also true for other, arguably more potentially dangerous and heavy, helmets including football helmet, welding helmet, lacrosse helmet, etc. Therefore it is safe to assume that bicycle helmets are allowed too.
Now, in terms of carry-on luggage allowance, I think you will be fine. For starters, the combined volume of your small backpack with the helmet attached will most probably be comprised within your maximum allowance. Moreover, I have flown with items strapped to my carry-on backpack multiple times and was never questioned. If worse comes to worse, either jam the helmet inside the backpack, wear it, hang it from your trousers and call it a keychain, or do something so that it does not appear to be part of your carry-on.
I have done this many times with a standard roll-away suitcase. I see no reason that a backpack would be different. I have always joked that if anyone hassles me, I will wear the helmet, and say I have fear of flying.