When filing jointly, you must include both SSNs. If your spouse will not share their SSN, then I would not file a joint return. Filing a joint return makes you jointly and equally liable for all facts, income, deductions and claims on the return. If your spouse won't share their SSN, I would worry about what else is going on and I would not want my finances tied up with them on a joint return.
When filing married filing separately, you need your spouse's SSN so the IRS can match your facts with theirs, since there are certain things you can and can't do when filing separately. If you don't have the SSN, you can't e-file. Leave the SSN blank, and print your return. Attach a statement explaining that your spouse refused to provide their SSN, and give as much identification about your spouse as you can (full name, birthdate, last known address). Sign your return and mail it to the IRS with your statement. You will also have to print and mail your state tax return.
