I was looking over the reference sheet inserts that come with the various expansion packs and saw an interesting line:

"Keep this sheet on hand so that players can reference this table at any time."

This is specifically in regard to the Maneuvers table for whichever ship is in the pack.

I have yet to do any official, organized matches. Do tourneys and such require that you have the reference sheets for all your ships on hand and available for your opponent to reference? Would I need to bring mine if I were to participate in a tournament at some point?

I'd plan to have them along anyway just because they're handy, but it still seems worth asking if I'd be required to bring them.

As an aside, how do tournaments handle players wanting to check on the available manuevers for their opponent's ships? Is asking to see the reference sheet allowed/commonly done?

You aren't required to have them, no. If you were, it would be included in the tournament rules.

If someone wants to see the maneuvers, all they have to do is ask to see the dial. It would be a poor sportsman indeed who turns down an opponent's request like that.

Some would also argue that you can't have reference sheets on your opponents ships with you either. Point them to the back page of the rulebook. You can have any maneuver reference material you wish.

No, you do not have to have any manuever referencing cards for tourneys.

I keep one copy of each one ship in my game bag, and will gladly let them see any reference material they may request. I think I've done it perhaps twice in all the time I've played.

If you look around online some peeps have made single card size versions and some for an A4 sheet.
Most squad builders will also print the dial or have the dial viewable in notes if you choose not to carry around a buttload of paperwork.

I myself was given a ref card by a friend, it has Imperial on one side and rebel on the other, is the same size as a pilot card and has all the ships up to wave4. I used it a lot until I memorized most of the dials.

I keep the maneuver chart "cheat sheet" that comes in the box in my case, in case an opponent asks for the maneuvers. I'll let them keep it until the end of the match. The only ones I don't have handy are the epic ships, the YT-1300, and the Firespray-31, since I don't want to cut up the rule books...