What cheeses work 'best' in melted cheese sandwich applications? I enjoy both toasted and grilled cheese sandwiches, but I generally only alternate between havarti and muenster cheese. I would like to branch out but don't know where to start. These are the factors I think are important in melted cheese sandwiches:


Melt well and fairly quickly
Fairly mild taste and texture
Generally available
Be a natural, dairy cheese (aka not Cheeze Whiz or American cheese) 


What cheeses fit these requirements? Are there other components that are vital to the cheese element of melted cheese sandwiches?
Gruyere is DELICIOUS. It melts without getting too gooey or soupy, and it is the traditional cheese component of a Croque Monsieur (if you're into that ham thing...). You could actually probably use any of the cheeses in that "variations" list, but I love Gruyere so that's my recommendation. It's widely available but tends to be kind of pricey here in the US.

Brie is also a good choice; since it is soft to begin with, it melts nicely. (Just make sure to cut off the rind before putting in the sandwich - that would be a weird texture combo.) It's great in grilled or toasted cheese sandwiches because of the richness and slightly buttery flavor.

As a local reference, check out Gorilla Cheese's menu for some tasty ideas; they do classic grilled cheeses with cheddar, American, gruyere or mozzarella, but they make others that have additions of other non-cheese components.