As my mother’s only child, do I have any rights toher estate?

First, you need to find out whether or not your mother had a Will.  If so, then if you are a beneficiary you will be notified.  Also, if your mother did have a Will, once it is filed and it has been filed you can go to the Probate Court in the county where she resided at the time of his death.  It would be a matter of public record and accordingly it can be viewed by anyone (for a small fee).  

Additionally, even if she had a Will, a parent may disinherit their child.  Children typically have no right to inherit anything from their parents.  Although, generally in such cases, there should be specific disinheritance language in the Will. Otherwise, if there is a Will and you are not made a bequest (gift) in it, you are unfortunately not entitled to anything.

Also, you should be aware some assets can be transferred outside of probate.  For example, if your mother had a small estate, property may have been transferred by affidavit or the like.  Additionally, some assets may have been held as "joint tenants with right of survivorship", in which case the other joint tenant would have received your mother's share of property by operation of law.  In addition, funds in an IRA, pension, 401(k), or other retirement plan bypass probate and go directly to named beneficiaries (unless the beneficiary named was her estate).  The same holds true for any life insurance proceeds (and as to such proceeds, unless you were a named beneficiary.

If your mother died without a Will, then she died "intestate".  Accordingly, the laws of the state in which she was domiciled at the time of her death would control.  Typically, an intestacy statute disrtibutes property in a situation such as yours - 1/3 to the surviving spouse and 2/3 the deceased's children.

To be sure of your specificrights under applicable state law you should consult with a probate attorney in the state in which your mother lived.