Now that my wife and I live apart, should we file a joint return or separate ones and both claim head of household? We have no children.

To file as "Head of Household" (HOH) you must have a qualifying child living with you in your home for which you have paid more than half the living expenses during the year so, based on your question, this does not apply to you. Failing that, you both may only file as married separately or married filing jointly. Which of those filing statuses to use depends on your specific circumstances.