My mom moved in with my husband and me in March. Can we claim her as a dependent come tax time?

The IRS law states that you must provide for more than 50% of what she needs to live. It's too late for last year, but it would be worth it to look it up on the IRS site or ask a CPA for this year. 

I think that her prescribed medicines will be considered medical expenses and a certain percentage of them be a deduction from her taxable income if she itemizes her deductions. I'm not sure about the incontinent supplies.

There are rules for claiming someone as a dependent. They have to make less than $4050 a year (2016) and you have to provide for at least half of their expenses. They have to live with you. You can't deduct for mileage unless you are a business and it is an expense. If you itemize deductions, you can include the medical things if there is enough of them to exceed the standard deduction. If you raised children, handling the deductions for your mother would be much the same except that she might also qualify for an increased elderly deduction.