This is one of the most frequently asked questions I hear during divorce consultations.  The answer is much more complicated than a simple yes or no.  There are several questions that need to be answered first:

1. Is the house community property or separate property?  If the house was purchased by either one of you before the date of the marriage, or if either one of you entered into the contract to purchase the house before marriage, then it is separate property.  If your spouse is the one who purchased it before marriage, you are almost certainly not going to be able to keep the house, no matter how long you have lived in it.  If you purchased the house during the marriage, then it is community property.  If the house is your separate property, you can almost certainly keep it (provided you can buy out any reimbursement claims your spouse may have).  If the house is community property, you may be able to keep it, depending on your answers to the following questions.
2. How much equity is in the house?  It is very important to know how much equity is in the house.  This is determined by how much the house is worth minus how much is owed on the mortgage.  If you purchased the house a year ago with very little down, then there is probably very little equity in the home.  If you purchased it 20 years ago, the equity is probably significant.
3. Can you afford to buy out your spouse’s share of the equity?  Most of the time, the community estate is going to be divided 50/50, so your spouse is probably entitled to 50% of the house.  That money does not necessarily have to come from the house, though.  For example, if the equity in the house is $100,000, you need to have $50,000 somewhere to put on your spouse’s side of the ledger.  Perhaps there is an investment account that has $50,000 in it that can be awarded to the spouse.  If there is not another $50,000 out there, can you cash out enough money on the refinance to buy out your spouse’s share?  If you cannot afford to buy our your spouse’s share of the equity one way or another, the odds are very slim you will be able to keep the house.
4. Can you refinance the mortgage into your own name?  In order to keep the house, if the mortgage is not solely in your name, you would have to be able to refinance it into your own name within a certain period of time.  I normally see anywhere from 90 days to 6 months after the date of divorce, although in certain rare situations I have seen parties agree to a longer period of time.  If your income is not sufficient to qualify for the refinance on your own, you will not be able to keep the house.  If you have been a stay-at-home parent, even if you recently started a job, you will most likely not have a long enough work history to qualify to refinance within the necessary amount of time.
5. Can you afford to pay the mortgage on your own?  I always recommend against including expected child support in the budget when trying to determine if you can afford the house.  Too many people fail to pay child support in a timely fashion, and the child support could be modified down the road for a variety of reasons.

In the majority of cases, I see divorcing couples selling the house.   Sometimes, parties sell the house because everyone wants a fresh start.  Sometimes parties sell the house based on the answers to the questions above.