The Geometric D type heads are the most common and so are the chasers for them. There are a few copies, mostly German, that take the same chasers. The German heads are just as good as the USA heads, but look a little bit different. The D heads come in 5/16, 9/16, 3/4, 1, 1.25 and other capacities. Each capacity head takes chasers that only fit that head. For instance, a 5/16 D head uses 5/16 D chasers, which are available in thread sizes up to 5/16" or 8 mm. Geometric made other heads, like the K, but the chasers are harder to find.
My first die head was a Geometric 9/16 D that had a 1900 patent date stamped on it. It is probably 100 years old now. It took brand new chasers and I made hundreds of parts with it. These things were made to last.
Look in a tool catalog (MSC, for instance) that lists chasers to see what is actually available new and what they cost. Genuine Geometric chasers are USA made, but I have a few good quality import versions that work fine and cost less. You can usually find lots of Geometric D chasers on eBay. Learn to recognize when they have been reground so many times that they are not worth buying. Yes, they can be sharpened with a surface grinder and a holding fixture.
There have been a number of other die head designs made over the last 120 years. They all worked fine when new, but most have become unpopular, resulting in difficulty finding chasers. I think there are a few companies that specialize in chasers for screw machine shops and probably have some of the less popular types.
I have seen a lot of eBay listings for die heads where the seller called it a "geometric," but it was actually one of the old designs that does not take Geometric D chasers. Most would be almost worthless. You need to study the pictures so you can tell what is being sold.
Larry