I live in Arkansas. My father died a few years ago. He had been married to my step-mother for 50 years. They had no children together and she has no other children. The only living relative she has is a sister and a niece and nephew. What are my rights to her estate in the event of her death?

None. Unless your step mother adopted you, you are not her descendant. Therefore her estate (unless she leaves a will) will go to her sister and her sister's children.

http://law.uark.edu/faculty/buehler/2012Spring/Arkansas-Guide-to-Inheritance.pdf

Was there a will? If not then you may still have a stake in the property.

This is something that a lawyer in that feild of expertise will have to look at and explain to you.

You say step mother. She was married to your father for 50 years. You don't mention your age. Did she ever legally adopt you?

Does she have a will?

Whatever is in her will, goes. Unless she legally adopted you, you won't be able to contest the will and win. IMO.

If she did adopt you and then left you out of the will, you could contest it claiming the rights of a child. It won't be quick, it will be costly, and you may lose, but you would have a legal right to contest.

If she has no will, and didn't adopt you, then her estate goes to her family.

Did your dad leave you anything when he passed?

As Alty asked, did your dad leave you anything when he passed? Did he have a will or did he die intestate? If he did not have a will, you should have inherited a portion of his estate. If he left a will, leaving everything to his wife, then the estate will be distributed per the terms of her will, or per Arkansas intestacy laws, which do not inherit a step-child that was not adopted.

You have no rights what so ever, unless you are named in a will.
I am assuming you were not adopted by her at any point.