Married filling separately vs Head of House

If you lived together anytime in the last 6 months of the year you can not file Head of Household.  Only Joint or MFS.  If you both agree then Joint is usually better.

If you all lived together for all of 2023 you cannot file as HOH and neither can your spouse.  In order for either of  you to file HOH you had to live apart for at least the last six months of 2023.

You can file jointly or you can file separate returns.  As for which one claims the children....you may want to seek legal advice for that since all of you lived in the same household.  

Or you can use the IRS tiebreaker rule:

https://itap1.for.irs.gov/owda/0/resource/Commentary_Files_Redirect_ITA/en-US/help/tbrk09.html

If you were legally married at the end of 2023 your filing choices are married filing jointly or married filing separately.

Married Filing Jointly is usually better, even if one spouse had little or no income. When you file a joint return, you and your spouse will get the married filing jointly standard deduction of $27,700 (+$1500 for each spouse 65 or older)  You are eligible for more credits including education credits, earned income credit, child and dependent care credit, and a larger income limit to receive the child tax credit. 

If you choose to file married filing separately, both spouses have to file the same way—either you both itemize or you both use standard deduction. Your tax rate will be higher than on a joint return.

 Some of the special rules for filing separately include: you cannot get earned income credit, education credits, adoption credits, or deductions for student loan interest. A higher percent of your Social Security benefits may be taxable. Your limit for SALT (state and local taxes and sales tax) will be only $5000 per spouse. In many cases you will not be able to take the child and dependent care credit. The amount you can contribute to a retirement account will be affected. If you live in a community property state, you will be required to provide additional information regarding your spouse’s income. ( Community property states:  AZ, CA, ID, LA, NV, NM, TX, WA, WI)

 If  you are using online TurboTax to prepare your returns, you will need to prepare two separate returns and pay twice since with online, you get one return per fee.

https://ttlc.intuit.com/questions/1894449-married-filing-jointly-vs-married-filing-separately

https://ttlc.intuit.com/questions/1901162-married-filing-separately-in-community-property-states

https://ttlc.intuit.com/questions/1894449-is-it-better-for-a-married-couple-to-file-jointly-or-separately