In the Season 2 finale, Sheldon is legitimately disappointed that no one came to his house to hear the Nobel Prize announcements, and Dr. Sturgis - the only other person invested in it - is unable to join in. Sheldon starts crying while his older self admits that at that moment, he felt that he would be alone forever.
After Isaac Asimov dies in "Babies, Lies and a Resplendent Cannoli". Sheldon is deeply affected by this, but Dr. Sturgis and Dr. Linkletter are the only ones who sympathize with him, and they quickly become more interested in hitting on Connie.
After he has to keep Georgie and Mandy's pregnancy a secret in "Uncle Sheldon and a Hormonal Firecracker". He is terrible at keeping secrets, and he knows this, but not even Mary will take his side.
In "Legalese and a Whole Hoo-Ha", he just wants to do science, but everyone else desperately wants to make money off of him, and so they obstruct him from doing science.
In "Ants On a Log and a Cheating Winker" one thing after the other goes wrong for Sheldon when he comes home, he has his room taken over by Georgie and Mandy without anyone in the family bothering to tell him until the very last minute or even take his side, has to sleep on the couch and deal with baby Cee Cee crying all night (which Mandy does nothing about and dismisses Sheldon's complaint) isn't allowed to use the bathroom in the morning because of a new schedule set up for the family, and comes back to his dorm to find it taken over by another person (although he does eventually try and make friends with him)
Mary, after Georgie and Mandy's pregnancy is exposed. She is forced to leave church (and by extension, her job) for something that was in no way her fault. Throughout the whole experience, Pastor Rob is the only person outside the Cooper family who supports her.
George Sr. has been this since at least the fourth season finale. He's forced to work at a job he hates, but doesn't know how to do anything else and is in danger of losing it (which he does towards the end of season five), ultimately resigning before they can fire him (though he gets re-hired in season six). He is frequently belittled by his mother-in-law, his wife and his children and is overall deeply unhappy with his life and feels unappreciated by everyone. No wonder he was driven into an early grave.
Georgie:
After Dale's store is robbed. He is racked with guilt over the incident and uses his own money to cover the amount that was stolen. Then he gets fired after Dale takes the money. Fortunately, this is reverted in the Season 4 premiere.
After he gets Mandy pregnant. Yes, it was wrong for him to lie about his age, but he understands that now, and he tries to do everything he can to support Mandy, but she rejects most of his offers. Meanwhile, his own family won't do anything to support him through this process, and they actively makes it worse in some ways (e.g. blabbing about the pregnancy to everyone in town).
Missy in "A Lot of Band-Aids and the Cooper Surrender". She is made fun of by her friends at school for her hairy legs, which only happened because Mary doesn't let her shave. Throughout this, Connie is the only one who has any sympathy for Missy.
Missy acts out and talks back to her parents, but its also clear she's only doing this because she feels like The Un-Favourite.
Connie is this in Season 5, with the storyline about her running a secret casino. She is incredibly rude to Georgie, even though he does everything he can to help her. However, it is clear from the beginning that Connie wanted to run the casino by herself, without having to work with Georgie (as well as the police union and June). Becomes this even more in the season six finale when her house is destroyed by a tornado and most of the money she made from the casino (which had been keeping at home following a recent robbery) has been lost in the storm.
Tam, Sheldon's best friend. His family was forced out of Vietnam and into the States after his father fought on "The Wrong Side", and his family's first attempt at a new life is dashed by racists. In Medford, it is implied that he has no friends besides Sheldon, and his home life is strict. Things get better for him though...
Billy Sparks. The Big Bang Theory set him up as a bully who enjoyed making Sheldon's life miserable. By contrast, this show portrays him as a dim, but friendly Cloudcuckoolander. He's clearly mentally handicapped, but everyone around him just dismisses him as an idiot rather than offer him help. He has no friends and his father left him alone with his mother, who seems embarrassed by him. Even Sheldon, who used to hang around with him in prior seasons, pretty much ignores him. Missy still hangs out with him from time to time, but it's mostly out of pity.
Dr. Sturgis. He's a social outcast, widely disliked by everyone other than Sheldon and Connie. Even the other Coopers are indifferent towards him. George tries to befriend him at Mary's behest, but it's clear that he only does so out of pity and obligation to his wife. He's also suffering from mental problems, breaking up with Connie because he doesn't want to burden her, even though they both deeply care about each other. And he's mistreated by her new boyfriend Dale. It's hard to see such a nice man go through all that undeserved crap.
Wayne becomes this after he discovers that his wife has been cheating on him. Their marriage seemed very happy and Wayne clearly adored Darlene. And not only did she cheat on him, but she's carrying the other man's child while Wayne has no children. To say he's torn up over it is an understatement.
Paige qualifies ever since her parents divorced. She starts acting out because she blames herself and can't handle the sudden changes. Even though she framed him for shoplifting (thankfully, no one was caught), Sheldon can't help but try to comfort her. Then when she goes to college, she is unable to make any friends and is not treated seriously by her colleagues. Then she reaches out to Sheldon, only to see that he has made friends in college and he acts, well, "Sheldon" about it. It gets so bad that she runs away for 3 days, and then drops out of college.
Dr. Linkletter in Seasons 4-5. He is very uncaring towards Sheldon, but he was only put in the position of looking after him because Dr. Sturgis left the university, and Linkletter has tried multiple times to get Sturgis to come back.
Dale in Season 5. He's a rude, sexist jerk most of the time. But he has no friends or family left, and it's clear that he is very lonely and has no idea what to do about it. Connie uses this to her advantage, pointing out the fact that he likely won't be able to find another girlfriend.
Mandy, after she gets pregnant. She coldly rejects all of Georgie's sincere offers to help her throughout the pregnancy, over the fact that he lied to her about his age, and the fact that he is overall less mature and responsible than her. However, no one else in her own family or the Cooper family (except Connie) does anything to support her at any point.
Mr. Gilford in "A Math Emergency", due to his half-assed philosophy of Screw Politeness, I'm a Senior!. As he puts it, "I'm old, I'm alone, I'm gonna die soon!" And he does in fact die at the end of the episode.
