How do I determine the cost basis for some XOM stock that my deceased husband inherited from his mother

Was there a valuation made at the time the stocks were transferred over? That would be your baseline value for the stocks.

The basis would be the stock price on the day the previous owner died.
If you need to prepare a return for your deceased husband, the basis would be the price on the day his mother died. If you inherited the stocks from your husband, your basis would be the price on the day he died.
You can look up historic stock prices here:
http://www.marketwatch.com/tools/quotes/historical.asp

Was the valuation made at the time the stocks were transferred to my husand and myself, or is it the valuation at the death of my husband and when they were transferred to my trust?

It would be both. For tax purposes there was the initial taxes when it was received. Then depending on how the stocks were held. There is a valuation done at that time. Mostly it has to do with state laws where you live as to how property is passed on between husband and wife. For example. If the stocks were held as separate property then all of it may be subject to taxation. But some states have laws saying its joint property so then only 1/2 of the value will be considered.

If all of this has passed through a lawyer via an estate (or trust) then they should be able to point you towards a qualified tax professional to guide you through.

Parent to child is different then spouse to spouse. That is why I recommend seeking a professional in your area.

Taxesforaliens is correct. When your late husband inherited the XOM stock from his mother his cost basis would be equal to the valuation of the stock value on the date of death of his mother (note - there is also a possibility that the executor of his mother's estate may have chosen an alternate valuation date of 6 months later, but this is typically done only if estate taxes are due - you can check with the executor of your mother in law's estate about that). Then when your husband died and you inherited the XOM stock, your cost basis is the valuation of the stock as of the date of his death.

I don't understand what you mean by "valuation at the time the stocks were transferred to my husband and myself," please clarify. Did your husband inherit the stock from his mother, or did you?