In the episode "Tricycle Thief," Susie accuses Angelica of stealing her tricycle. Tommy tries to give Angelica the benefit of the doubt, and then says to look at Angelica's hands, since they'd have paint on them if she opened Susie's (recently painted) shed to steal the trike. Angelica's hands do indeed have red paint on them, just like Susie's shed. This becomes the "final nail in the coffee," and even Tommy no longer believes Angelica is possibly innocent... because she was literally caught red-handed. Turns out she actually didn't do it, though.
Also, during that same episode, Susie tries to console Chuckie by saying "Angelica let go of the balloon" (which was tied to Cynthia's leg, as Susie's way of punishing Angelica for stealing her tricycle). She says "Angelica let go of the balloon the minute she opened my daddy's shed and stole my trike." As she says this, she gestures to the shed, and then to "her" tricycle (the one Angelica claimed was her own). The camera pans from the shed to Angelica's tricycle, and between the two, you can actually see that Susie's tricycle is underneath her porch. None of the characters notice right away.
In "Chuckie's Wonderful Life", Chuckie is told by his guardian angel that he's such a good friend that he gives Tommy the bravery to stand up to Angelica and other bullies. Go back a season to "Rebel Without a Teddy Bear", where Angelica is loudly coercing Tommy into throwing his mother's necklace into a garbage can. Chuckie actually stands up to ANGELICA in order to keep Tommy from doing it.
Angelica's parents spoil her and while Drew has a backbone at times, he still gives in. He finally punishes her in "Angelica Runs Away" and Angelica is seen being punished more often after this. It serves as a Crowning Moment of Awesome and Character Development for Drew.
In the episode "The Dog Broomer" a subtle hint she was a fraud was her mentioning she was from various cities. At first, she says she's from Helsinki, then she says she was from Stockholm.
The size of the characters' houses and yards change in some episodes. This is because it's seen from a baby's point of view, and they see things differently on their adventures.
There's a subtle arc with Stu in the first season. In the second episode he meets the owner of Mucklehunny toys who likes his inventions and wants Stu to make toys for him. He also is working on a Patty Pants doll in which he finally perfects for Mucklehunny in "Stumaker's Elves."
Chuckie calling a Jack in the Box a Jerk in the Box is not just the babies mispronouncing words, it's a Freudian Slip for Chuckie's fear and hatred of clowns.
In The Movie and Rugrats in Paris Aunt Miriam has brown hair while she has grey hair in the show. In "A Visit From Aunt Miriam" it's established her hair is a wig, so it's a different wig in the same hairstyle.
Angelica being jealous of Susie in "Doctor Susie" if you remember that she had previously thought of herself as a doctor in "Rhinoceritis".
When Grandpa is dreaming in the camping episode he says "Dust Bowl, Shmust Bowl, I'm not moving to California." It's been established they live in California and that the Pickles aren't originally from there, so Grandpa was remembering other than going on a tangent.
In "A Visit From Lipschitz" after meeting the Pickles family he decides to change his style of children's psychology. He also quotes Sigmund Freud, who in real life is a case of History Marches On due to new evidence, and it seems Lipschitz himself realized that after personal interaction with the babies.
The episode "Chuckie's Wonderful Life" shows that without Chuckie, Chaz is a mess, living alone in his house surrounded by empty pizza boxes and talking to a sock puppet. It gets worse when you realize that his wife (Chuckie's mother) died of an unmentioned disease (according to the Mother's Day special episode). And since Chaz doesn't have a kid, he wouldn't be as close to the other parents and therefore would have no friends to help him through his wife's illness and his own mental breakdown.
All Grown Up (thankfully) fixed this, but imagine somebody as borderline sociopathic as Angelica as a teenager or even an adult.
Calling Angelica sociopathic, even borderline, goes a bit too far, as she is only three. But yes, it would be terrifying to imagine how a child like Angelica, who receives little to no discipline whatsoever, would be as a teen or adult.
There actually was an episode where Tommy was kidnapped. The kidnappers returned him by the end of the episode, though.
In the episode "Toy Palace" there is a real, working time machine, set up as a Chekhov's Gun for the episode. In the Rugrats universe, not only is time travel possible, it's meant as a toy for children.
Fridge Brilliance leading off of this: the existence of time travel explains all the gross negligence on the parent's part, as well as explaining away the Comicbook Time nature of the show.
From the "New Kid in Town" episode, Josh's game "Pillow" involves him attempting to jump on the babies from the swings. If Angelica hadn't stepped in when she did, the little bastard could've killed them.