Hi Hollo
usually those considered at risk of multiples/ ectopic pregnancies or previous problems with bleeding / IVF /Clomid babies have early 6-7 wks scans (call them heartbeat scans in AC circles) - or to check the embryo is where is should be and viable.
Though some others have them too - like those not sure of their dates - for them it is called a dating scan.

The approx. 12 week scan is Nuchal Translucency / measures neck skin fold to check for genetic abnormalities like you mentioned etc.

You can ask for this scan at any age - and many do even if younger or those not even considering termination if a problem is found - just to know everything is okay is a great peace of mind.

the 18wk scan is morphology scan that most people have these days