In bughouse, a bunch of normal openings are instantly unsound because they expose squares for pawn drops. See the Modern and Sicilian below:
Some principles that might lead to useful openings:
It’s typical not to castle.
It’s unwise for either side to push his c-pawn or leave his f-pawn vulnerable.
At high levels, white plays offense while black defends and trades.
Knights are more valuable than bishops.
A brief search shows me that 1. e4 is considered strong (and that it’s definitively the strongest opener in crazyhouse). It’s hard to come up with long opening lines because piece drops lead to an explosion in possible variations, but opening structures like this seem respected:
The most common black defense is apparently 1. e4 e6 2. d4 d6, but I suspect the most powerful response is an Alekhine’s Defense sacrifice with 1.e4 Nf6 2. e5 d5! followed by a very secure environment if black castles and drops a pawn on h6.
But isn't gifting a horsey to the diagonal opponent considered harmful? At least in the early stages of the game.