Here is an overview of all the grandparents in Single Again and their relationship with grandchildren and children:
Frank's parents, or Abby's “grandparents”: Relatively little is known about the grandfather so far, only that he got a promotion that forced his family to move. This led to Frank meeting Jess and Shannon and is essential to the plot. Apart from that, the only other mention of him is that his wife and he sold the house and gave the money to Frank and Jess. Frank's mother obviously had quite a good relationship with Jess (Abby's middle name is Frank's mother's first name) and she was obviously pleased that Frank finally asked Jess the one question at the time. It is highly likely that they are both still alive.
Jess's parents, or Abby's other grandparents: Disowned Jess when she got pregnant and didn't want to have an abortion. Even though we have no clear confirmation, I assume that they (at least the grandmother) would like to put the whole thing behind them. How far that goes and whether she would like to meet her grandchild is unclear. I can understand Frank and Jess's decision to tell Abby that these grandparents are dead, and I hope Abby gets another chance to tell them (or at least her grandmother, if the grandfather is already dead) to their faces that the only grandparents she has and are alive are the ones she has on Frank's side. Preferably with a clear “As far as I'm concerned, I see it like my parents and you're dead to me.”
Shannon's parents, or rather the grandparents of Lilly and Becca: Nothing is known about the grandfather so far and he was never mentioned, which could be an indication that he is already deceased. The grandmother is or was a rather old-fashioned woman and raised Shannon strictly (according to Becca), which obviously backfired badly. She seems to have a good relationship with Becca and Lilly though (Lilly refers to her as “Abu”, which I think is a Spanish version of granny, while Becca would like to have her taco recipe, neither of which would probably be the case if they didn't at least have a neutral relationship with her). Since Abby also knows Grandma Oxton's tacos, I assume Abby and she also have at least a neutral relationship with. That being said, I think that if they didn't have a good relationship, she would probably automatically like Becca's tacos better.
Now the following is not to bring up the “Becca is Frank's daughter” discussion, but I think if Shannon's pregnancy was the reason she had to move away at the time, but she will probably assume that Frank will be Becca's father since Frank was Shannon's boyfriend at the time. Even if she's wrong, she'll probably still assume it and therefore probably have been more open with Frank and Abby because the family finally looked more or less as it should have looked, in her eyes. She will have accepted Abby's existence as she may consider herself the guilty party in the scenario and Frank could not have known about his, in her assumption, daughter.
Final conclusion: I think Frank's parents are by far the most likeable grandparents. And even if we only get hints of it here and there, I strongly suspect that Frank's parents have been waiting years for him and Jess to finally get married.
Jess's parents: by far the least likeable. Frank is rightly angry with his ex-parents-in-law, who I get the feeling either don't realize the consequences of their actions or dismiss it as a small thing, like they forgot to water the flowers while Jess and Frank were on vacation.
Grandma Oxton: I'm still unclear about her. We only have a handful of lines about her in the plot and accordingly, I can only make assumptions about her and her mindset. I can either make her worse than she really is, or unintentionally make her better than she deserves, so I'm neutral on her so far.