"Can one spouse claim married filing separate and the other file head of household "
Yes, but ONLY if the spouses have lived apart for the entire last 6 months of the year (from July 1 to the end of the year). If you live together for even one day during that time, your choices are limited to married filing jointly, or married filing separately.
Head of household is only really intended for single parents, meaning you are unmarried, or have a long-term separation from your spouse. HOH does not mean "head of the family."
