There are as many different kinds of fatties as there are opinions. I like to use almost any kind of sausage I can get either in patties or otherwise outside of casings. I've used lamb sausage to make Greek fatties, Italian sausage to make, well, Italian fatties, breakfast sausage is good for almost anythinng, ground beef was good for bacon cheeseburger fatties, SQWIB used chip steak. It doesn't matter because it all depends on what you're going for.
But keep looking around for inspiration. After seeing a post here, I came up with the idea of doing a Greek fatty the other day, lamb sausage filled with feta, calamata olives, stuffed with garlic, caper berries and rosemary. It went in the freezer, so we haven't eaten it yet. After looking at SQWIB's post above (very nice, BTW!), I might try wrapping philo dough around one and baking it to finish it off. Served with a little tzatziki, yumm!
It's all good!