In one episode Joey goes to amateur night at a comedy club, but gets interrupted by Phyllis Diller who takes his place. It may seem like a random guest star until you know that Diller lived in San Francisco for part of her adult life.
When DJ gets her first car, Danny plays the song "The Horn Went Beep Beep Beep" on the guitar. It may look like Danny looking like a dork, but in a previous episode he had an attachment to his old car and the song had some meaning to him.
Danny being able to buy a new car in a season one episode, renovate the entire basement twice, as well as take vacations (when his roommates don't pay rent and he lives on a local TV newscaster's salary) makes more sense if his late wife's life insurance paid for it.
Also, being a popular local sportscaster (and later morning show host) in a large media market such as San Francisco suggests that Danny might be fairly well-paid as well.
Becky announces she's pregnant in the season four finale, which is set in May. It's also established she has the twins in November. This might seem like a standard TV pregnancy, but her and Jesse got married on Valentine's Day in February. The twins were conceived then.
Why is Danny obsess with cleaning when in earlier episode he is very reluctant to do so? It is because in that same episode Danny state Pam took care of that and so the reason why he go out of his way for cleaning, it is his way of recreating being with his wife.
There's a theory men are attracted to women who remind them of their mothers, this is why Danny dates Gia's mom in the last season. She's divorced, a single mom, and her name is Clare just like Danny's mother.
In a first season episode Joey gets the chicken pox as an adult and says he never got them as a kid. In the episode when he reunites with his dad, his dad tells him he had the chicken pox as a kid and he watched cartoons with him until he got better. It's possible to get them again if you had them at a very young age. Joey's dad is an authoritarian, which Joey resents and it's possible it happened when he was too young to remember it, hence why he doesn't remember them watching Popeye together.
Jesse's outburst when he thinks that DJ has started drinking beer makes more sense when we find out that DJ's mother, Jesse's sister, died in a car accident because of a drunk driver.
Jesse's name is a Meaningful Name. Jesse is an avid Elvis fan, and what was the name of Elvis's twin brother that died in child birth? Jesse. He probably saw it as a sign as a child.
When Joey's new girlfriend senses that Joey isn't happy with the way Jesse always takes the lead and makes decisions in their friendship and partnership, she encourages him to stand up for himself. This results in a huge fight between the two, in which Jesse outright refers to the woman as "Yoko". While the story of Yoko Oh No is well known, remember that Jesse is also a rabid music fan. It makes perfect sense that he'd draw such a comparison.
Granted, Michelle (especially in the later seasons) is usually annoying and bratty, but there was at least one scenario where she was potentially Lethally Stupid. In a later episode, she decides to take up a hobby. Fair enough, but the hobby in question was collecting leaves, and if that wasn't irritating enough, the leaves left on Stephanie's bed were the ones with little bugs on them. Did Michelle essentially expose her sister, herself and everyone else in that enabling, dim family to Lyme Disease?!
More like frigde tearjerker: In 1x13 DJ tells Jesse and Joey that she is so sick of sisters. Jesse tells her that one of the benefits of being a big sister is having two younger sisters that love her, need her and look up to her. Coming from someone that lost his own big sister, this takes on a deeper and sadder meaning.
Then there is the scene in 3x08 where Jesse tells the Stephanie and DJ that the 5 year age difference between them seems big at the moment, it won´t matter when they are older. Given the fact that Jesse was about 5 years younger than Pam, then it sounds like he is basing it on his own experience. Thinking about it from ankle highlights how much Pam meant to him, and how much he lost when she died. Having these conversations with his deceased sister´s daughters about relationships between siblings and how great it is to have a sister, must be painful for him.