I will be travelling on a Boeing 777-300ER and want to use my Sony headphone which has a one-prong connection. Would I need an adapter for a two-prong situation? My return flight is on a Boeing 777-200.
It's hard to say for certain if you need an adapter. Personally, it has been over a decade since I have been on an aircraft that required the two-prong adapters, and I have flown (in that time) on Air Canada, WestJet, United, Continental and British Airways in various types of aircraft.
Given how inexpensive an adapter is (a couple of dollars on eBay), if you have time, I'd pick one up and keep it in your headphone case. Many headphones come with these adapters - my noise-reducing headphones come with a pair of them to adapt them to the two-prong jacks you describe, as well as the older 3/8" headphone jack.
My recent experience has been on 2-3 Asia based carriers with the 2 prong jacks.
However, my single prong headphones worked perfectly fine. It just doesn't fit very well and the contacts are slightly off so I'd have to set the jack in a specific location. The headphones did come with an adapter but it's long been lost.
It is not as simple as just buying an adapter since Cathay Pacific business class UK to HK use a twin socket with two different diameter connectors. My Bose adapter with two 3.5mm plugs did not fit. I think the thinner socket with two contacts is 2.5mm and the thicker one is 3.5mm with three contacts. I haven’t been able to find an adapter like that and I don’t know how they are wired to make one up myself.
I was in China last October. I flew business class on Cathay Pacific between US and Hong Kong. They had 2-prong plug for the Bose headphones they gave out. I unfortunately didn't have a plug for my own Bose. So I am going to buy my own plug for my next trip so I don't have to use public headphones.