A. This is a perfect example of why parenting is both an art and a craft. At any given moment, you may be called upon to, say, explain to your child why people have wars -- and to know which bathroom to head for if she really has to go. From the lofty to the mundane, in less than five minutes.
Of course, it's smart to have her try to pee before you leave the house -- even if she says she doesn't need to. If she goes then, she may not have to go again until she gets back home.
But if the urge does strike while you're out, sending a child of 3 into the ladies' room unattended isn't an option -- she's too young and will be for another two or three years. Luckily, although she may seem like a big girl to you, she's still small and toddlerish enough to take with you into the men's room. When your only choice is to take her into the gents', just pick her up and whisk her in -- chances are neither she nor the rest-room patrons will mind. At this age, most kids are very matter-of-fact about bathroom business. If you're hesitant because men's rooms tend to be on the unsavory side or because your daughter objects to going in one, then your options are few.
If you're in a store or restaurant, ask whether there's a wheelchair- accessible or family rest room you could use. These are often found in a different area from the standard multistall rest rooms, and their single-toilet design allows you to go without regard for what's posted on the door. Or look for a smaller store or restaurant -- it's more likely to have either only one rest room or a men's and women's room with just one toilet (for employees' use but sometimes offered to patrons). And make it clear that it's your daughter who needs the facilities, not you (play that pity card when you have to).