How does one go about finding assumable mortgages in their area?

What is it you are trying to do. The only time you assume a mortgage is if someone is selling.

Exactly. I would like to get into a home and take over payments, thus assume their mortgage. But I was wondering where to look to find such availabilities.

I doubt if you can find such specifically advertised. You might try to check foreclosure lists or look at forsale by owner situations and ask whether you can work out an assumption of their loan.

How does one go about finding assumable mortgages in their area?
This is what someone is doing with us. We have a mortgage, but bought a new home and need to move, I think its called a wrap around loan. So I am on the other end of this. I have someone taking over myhome payments. I am so scared on my end and just started investagating it

There was, in the very recent past, mortgage loans that were "assumable" which meant that you simply paid the seller's equity and "assumed" the seller's mortgage loan from the point where the seller exited any further responsibility and you, the buyer assumed responsibility. There were advantages such as 1) no need for the purchaser to qualify for a mortgage loan 2) If time was an issue for the seller or buyer, it was very quick to arrange for an atty or title company to draw up the docs, etc, meet for a closing and the deal was consummated. I don't believe many of these types of mortgages loans remain as when I assumed one in 1993 I was advised by the seller, the title company that few remained at that time. I am not familiar with the "wrap around" loan metioned in a previous answer.