The "Getting Out" post below had a lot of responses that mentioned putting a car seat on top of a grocery cart. While people do this all of the time, it's really, really not safe. Here's a blog entry that describes all of the reasons, complete with illustrations:
https://treadingragingwaters.blogspot.com/2012/06/shopping-with-baby.html?m=1
Still don't believe it isn't safe? Check out the news story about a 3 month old baby who died when the car seat fell off of the shopping cart:
https://macon.13wmaz.com/news/news/58804-infant-dies-after-falling-out-shopping-cart
I'll be honest - until a friend sent me this information, I had no idea it was unsafe and used to do it, too. 
Alternatives?
-Put one car seat in the basket and wear the other baby
- Use a stroller and pull the cart (what I usually did until we could do double carts)
- Use the stroller and put all of your stuff in the basket or in shopping bags that you carry (good for quick trips). 
 
Re: PSA: Car seats and shopping carts
My carseats don't even fit the top of the cart, but I didn't really know it was a no-no either until I was on TB. The sad thing is I see it at least 2-3 times every time I go out shopping. I used to go shopping with my mom or DH and we would each push a cart with the seat inside. Now I either use the double stroller and put all my stuff in the basket or use one of the double seat shopping carts.
Peanut Butter and Jelly!
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Our twin boys arrived at 36w5d due to IUGR and a growth discordance
Be kinder than necessary, for everyone you meet is fighting some kind of battle.
I honestly had no idea until I saw this mentioned here on TB. We used to do it all the time with DS1, thank goodness he was never hurt.... I am definitely never going to do it with the babies, I will either push and pull the stroller and cart, or wear one baby and put the other seat in the basket. Sooo scary.
Oh and what's really scary? Our local Target has the new shopping carts that you really can't put a carseat onto.... I see parents balancing the car seat perpendicular on the sides of the basket out in front. Waaaay dangerous!
I didn't know this, but we hadn't tried it yet. Thanks for posting.
We have been wearing them in Bjorn's for shopping trips so far.
Boy/girl twins born at 37w1d and 37w2d