Altium: How to hide rats nests?

In Altium Designer 16.x, while viewing the PCB go to View > Connections > Hide All.

This will hide all of the ratnest nets. If you select a component that one component's nets will be shown as a convenience, but other than that I think this solves your problem. I find it useful to enable this option during initial placement of a crowded design.

You can do this by going to your layer set properties (click the colored box at the bottom left of your screen):

On the right-hand side uncheck the box labeled "Show" next to "Default color for new nets":

This will hide all airwires at once.

Right click on a net, and choose Net Actions >> Hide Nets.