This is mostly for PedNurse, but also I hope anyone else lurking who has gone through recurrent loss might benefit from others' experiences. Obviously I'm not a doctor, and I'm sure this is not going to be a full list, but this is what I remember. Anyone else who has been through testing, please feel free to comment and add in. Again, this is not medical advice as I am not a doctor, this list is just meant to be a place to start a conversation with your doctor if you are not sure what you can ask for (although hopefully your doctor should know...:))
PedNurse, I completely agree with Bonita that I don't see why you should have to wait for "next time" for most of these, and that I hope your doctor is open to and knowledgeable about different causes of recurrent miscarriage.
1. Genetic - As you mentioned, there are different types of genetic screenings. I don't know what all of them are, but the one that I know of that may possible cause recurrent loss is a MTHFR mutation. I have this. The thing that it does is make it harder for your body to process folic acid, which is really important to developing new tissue including new babies. The treatment for this is pretty benign - extra folic acid and vitB. I'm sure there are all sorts of other genetic tests they do, I just don't know what they are.
2. Structural - Not sure if they can do this one while you're actually pregnant, but at some point they can do what is called a Hysterosalpingogram (HSG) which is basically where they shoot radioactive dye into your uterus, take an MRI, and look for any sorts of structural abnormalities that could be causing problems.
3. Blood tests - They can check your thyroid levels (thyroid problems can wreak havoc with pregnancy), as well as possible blood clotting issues that may be treated with baby aspirin or possibly low molecular weight heparin.
4. Betas - Testing for progesterone levels is a big one. Low progesterone seems to be a common theme in miscarriages, and is easy to fix with either an oral pill or suppository.
5. Bacterial - There are certain types of bacteria that can live on your cervix that seem to be implicated in recurrent loss. If found, these can be treated with antibiotics.
Those are all of the ones that I can think of, but again, your doctor should know a heck of a lot more than I do...lol The good news is that almost all of these issues are treatable (genetic issues aren't "curable", as we can't change our genes, but may be treatable as in my case where I simply need to take a buttload more folic acid.)
I hope this is helpful, and anyone else out there that has been through testing, please feel free to add/correct and otherwise chime in.