Taking your cat to the vet
Cats are territorial creatures and rarely ever leave their home; when they do it usually means they're going to one of two places - the vet or the cattery.
Therefore when a cat is put into a cage they assume they're going somewhere "bad", and travelling becomes stressful. The cage becomes a fear-inducing stimulus, putting them on edge before anything has even happened. Because of this, you should practice habituating your cat to the cage, as a step to making vet visits and travel less scary.
When selecting a cage, you should choose one that is covered in, has a front door and the ability to come apart in the middle. It should be big enough for your cat to stand up and turn around in, but also small enough for them to feel nice and cosy while inside. Make sure the bottom of the cage is lined with a familiar smelling blanket and if you have a commonly used wire-top cage simply covering the cage with a towel will go a long way to making your cat feel more secure.
The scent of a cat's territory is very reassuring, and any major change to this can be distressing. Feliway is a synthetic version of the facial pheromone secreted by cats which can be used as a "spot treatment" on blankets or bedding inside the cage to send a message to the cat that it is safe and secure.
Time your arrival at the clinic just prior to your appointment, to avoid your cat having to wait for too long. If there are dogs in the room sit as far away from them as possible. Have your cat on the seat beside you rather than on the floor, so they are out of dog-sniffing reach.
Once you've entered the consult room, let your cat come out of the cage on their own - place the cage on the floor or table and open the door, giving them the opportunity to come out and investigate while you are chatting to your vet. If your cat does need a helping hand to come out, don't "drag" or tip them out the door. Take the top half of the cage off so you can lift them out gently or if your cage doesn't come apart, reach in behind your cat and support them underneath while lifting them forward.
Once your cat is out of the cage and ready for examination hide the cage so your cat can't see it, or they might try to constantly escape back in. Cats don't like to feel like they are "losing control" - give them the blanket from the cage to sit on while being examined or weighed; rather than slipping around on the table-top.
As soon as the examination is complete, let your cat back into the cage - although they might not like going in there at home, while at the vet clinic the cage is often seen as the cat's safe haven.
See also Teaching cats to love their carriers