If Lily supports her children and will not file jointly with her spouse, what is her filing status?

Lily's filing status as "Married Filing Separately" is appropriate under the conditions described. This status is typically chosen by individuals who are married but decide to file their tax returns separately from their spouse for various reasons, such as wanting to maintain separate financial responsibilities or in situations where it may yield a lower overall tax liability.

For Lily, since she supports her children and has indicated that she will not file jointly with her spouse, "Married Filing Separately" accurately reflects her situation. Furthermore, to qualify as Head of Household, one must be unmarried or considered unmarried on the last day of the year, and in this case, since Lily is still married but chooses not to file jointly, she would not qualify for that status.

While "Single" could seem relevant, it does not apply here since Lily is married. "Married Filing Jointly" is not an option she is considering, as explicitly stated in the question. Therefore, "Married Filing Separately" aligns perfectly with her circumstances and decision not to file jointly.