Clearly a capital s, and the separation of the words through color only makes it clearer. Perhaps they are knowingly using the name as some kind of brand-strengthening pun, but as I said in my last post, the second meaning is the correct one.
I disagree.
Let's break it down.
"Pets Mart" -- A mart where you can buy pets. Not really the case.
"Pet's Mart" -- A mart for pets. This seems like the most accurate title to me.
"Pet Smart" -- Doesn't even make sense. This is clearly not the store's name.
Now let's investigate the logos.

This older logo is a great example of Sploops' Bouncing-Ball-Apostrophe Theory. The "s" is lowercase and the bouncing ball functions as an apostrophe. So clearly, in this case, the name of the store is "Pet's Mart."

Now this newer logo keeps the bouncing ball apostrophe, but changes the lowercase "s" to an uppercase "S" and changes the logo colors to
PET'
SMART. If I were colorblind, I would definitely read the store title as Pet's Mart. But it seems the color discrepancy has made some of you think the title is Pet Smart (ridiculous). And honestly, the color discrepancy is all you have to make that argument.
Why would a corporation change their name from Pet's Mart to the nonsensical Pet Smart? The answer is they wouldn't. It seems to me that they are just using brand innovation to appease pet owners who really like puns. I bet when the CEOs picked the name Pet's Mart, they had no idea of the potential grammatical double meaning, but then some uppity VP had to go and point out the pun late in the day at some quarterly meeting. "Hello everyone. We think Pet's Mart would be really Pet Smart if we stopped abusing animals, HA HA HA!" At that point, the CEOs knew they were stuck with this awful pun for the rest of their lives. But the truck drivers and cashiers and janitors know that it's still
Pet's Mart.