Against an all-court player, you start your search for a weakness without any conspicuous hints. An all-court player is comfortable on every part of the court: baseline, transition, and net; offense and defense. An all-court player is a generalist, good at everything.
Being a generalist is usually advantageous. Koalas can't live anywhere that eucalyptus trees don't grow. Rats (no offense, all-courters), who will eat just about anything, are global. One food rats won't eat, though, is eucalyptus leaves. So, if you're facing an all-court player, feed him eucalyptus leaves: challenge his comprehensiveness with your specialty.
When you think of the great all-court players, like Pete Sampras and Martina Hingis, it's useful to remember who was able to beat them: highly accomplished specialists, like Patrick Rafter, one of the last pure serve-and-volleyers, or the new generation of power baseliners, led by the Williams sisters, whose sheer pace was too much for Hingis. The great all-courters rarely lose to other all-court players: they lose to players who can't do nearly as many things well, but can do one thing significantly better.
If you have a specialized weapon, an all-court opponent probably won't blow you out of the point before you get a chance to try it. The one thing most all-court players don't have is a single huge weapon. (Few have a Sampras serve.) You don't have to fear any one shot from an all-courter, so you have a little more time and a little more freedom to play your own game. If you're a power hitter, you can wait a shot or two before going for the winner. If your big weapon is heavy topspin, you can kick the ball up on whichever side of your opponent (probably her backhand) has more trouble with high balls, and you don't need to worry too much if she runs around it. If consistency is your strength, and you can retrieve, lob, and pass well, you don't have to fear too many brilliant shots you simply can't touch.
All-court talent doesn't always come with matching stamina. Very often the most talented players don't have to work as hard, so they're vulnerable to being worn out. Make your all-court opponent run as much as possible (Agassi style). A tired opponent will start to make more errors.
As against any opponent, you should test your all-court opponent on every type of shot you can send him. He may be comfortable on every part of the court, but that doesn't mean he's comfortable with every kind of ball. See how he handles a low, wide slice to his forehand at the baseline or a lob over his backhand shoulder when he's at the net. He's probably used to playing baseliners, so test his passing shots. Sometimes a player who loves to attack the net will panic a little when his opponent gets there first.
How do you handle your all-court opponents? Let us know at the tennis forum.