What to do if an unmarried couple have a mortgage together on a house and now want to separate?

Assets are divided however they like or decide to--if you are asking what a court would likely conclude, in the event the couple cannot agree and they litigate (or the couple simply want guidance), the answer is that

1) Each person would take with them any assets they brought into the living arrangment;

2) Each person would solely own anything he or she could prove he or she bought entirely with his or her own money and did not gift, in whole or in part, to the other;

3) For joint purchases or assets, they would likely each have a share of the value proportionate to what he or she paid for it.

The home can be sold by the two of them--all that is required is their agreement to sell it. Or one can buy it from the other, including at favorable  terms. The mortgage cannot be assumed (taken over) by either, unless the bank allows it--more likely, the one buying the home would need to refinance in his/her sole name.

For the house and mortgage, a real estate lawyer would be best; the same attorney can most likely handle any matters that arise with other assets, as long as matters are not too contentious. If there is significant conflict, a family law attorney might be a better choice to help.