Opaque flights, in the US at least, are far less common than opaque hotels - however they do still exist.
The most common companies that do them are the same as for hotels - Pricelist ("Name your own price flights") and Hotwire ("Hot Rates"). Some other websites do them as well, including Expedia and Kayak, however they normally don't have a separate interface for these, and only show them on occasion after you've searched for a normal flight.
Just like opaque hotels, there are disadvantages of opaque flights. Not only will you not know the airline you'll be flight (which could lead to extra fees like baggage fees depending on which airline you get!), but you also won't know the exact time that you'll depart or what cities you'll connect in - instead you'll be given a range of times (eg, depart between midday and 2pm) and a number of stops (eg, 1 stop). You will also generally not earn frequently flyer miles/etc.
There's a good description of why these fares exist in the answer to this question.