I was playing Commander with a couple of buddies the other day. One of them had a 10/10 creature and Broken Fall enchantment in play to regenerate said creature. One of the other players cast a destroy target enchantment instant on Broken Fall. The player with Broken Falls responded by putting it back into his hand, activating the text for regeneration and putting that regeneration onto the 10/10 creature. If that was played correctly by the Broken Fall player, how then do you ever get rid of the Broken Fall enchantment when he can just protect it by putting it back into his hand, still protecting his creature and then just replaying it again on his turn?
You have two separate goals here: Removing the enchantment, and removing the creature.
Removing the creature is the easier of the two problems
Regeneration only protects against lethal damage and 'destroy' effects. That means that a creature with a regeneration shield can be removed by -X/-X effects, exile effects, forced sacrifice effects, Pacifism effects, and many, many more.
Finally, regeneration only works once, so a second kill effect will work if you can play it before Broken Fall is played again.
Removing Broken Fall permanently is harder
When an activated ability is activated, all costs are paid and then the ability is placed on the stack. This gives opponents a chance to respond to the ability - but the cost cannot be responded to. This makes permanents that bounce themselves as part of a cost very difficult to deal with, as you can't target them in reaction to their activation. (Even if you could, they could just activate it a second time, and still escape)
The only way to target Broken Fall without it escaping back into their hand is to do so when they can't activate its ability, which is a very narrow set of circumstances: 1) There are no legal targets on the battlefield - that is there are no creatures at all, or all the creatures present have hexproof (and are controlled by opponents) or shroud. 2) The activated ability has been blocked (for example by Pithing Needle) 3) Split Second is in effect and nothing can be placed on the stack. Of course, Split Second also keeps you from playing anything, so unless the split second card you played was Krosan Grip, its not going to be very helpful.
There is at least one more way to deal with Broken Fall on the battlefield - create a board state where your opponent would rather let it die than save it. This is of course very situational and tricky to do (especially as it depends on your opponent), but can be accomplished by cards such as Doom Foretold. If your opponent has to chose between sacrificing their 10/10 or sacrificing Broken Fall, then they might decide that it is better to lose Broken Fall than to lose their creature. Of course, the downside of this strategy is it only works if Broken Fall is the least valuable part of their board!
And there is, of course, the ultimate work around: Mindslaver them, and then do whatever you want while they are enslaved!
It's tricky to be sure. One way to get around the ability is to cast a destruction spell with Split Second. While a spell with Split Second is on the stack, players cannot cast spells or activate abilities (except mana abilities).
You can also use a card like Pithing Needle to shut down Broken Fall's ability.
It's going to be tough. As long as there is a targetable creature on the battlefield, your opponent will be able to activate Broken Fall, sending it back to his hand. And since the enchantment returning to the hand is part of the cost and not the effect, you can't even respond in between Broken Fall being activated and the card returning to the opponent's hand. Basically, almost anytime something is "about to" happen to Broken Fall, it can be sent back to the hand.
So, getting rid of it from the battlefield will be very tricky. One of the only ways to do it will be to use a spell with Split Second, which prevents an opponent from responding before the spell resolves (see Krosan Grip). Other cards which limit activated abilities will also be very useful (see Pithing Needle). Although getting rid of Broken Fall from play is tough, you can also get rid of it when it's not on the battlefield. You might have better luck getting rid of Broken Fall from the opponent's hand, by forcing them to discard it. Alternatively, you could counter the spell next time they play it, preventing it from ever reaching the battlefield.
One thing you could do to kill the 10/10 creature is to have two creature-killing spells. When you put the first one on the stack, your opponent can respond by activating Broken Fall, which will grant regeneration when it resolves. You now have the option to play your second creature killer either before Broken Fall resolves, killing the creature before it gets regeneration, or playing it after Broken Fall resolves, killing the creature after it's been regenerated from the first attempt to kill it. Broken Fall can effectively only save the creature once on your turn (unless it's somehow put onto the battlefield again), so you just need to kill it twice.