Angst? What Angst?: Sarah Jane is a Technical Pacifist who always tries to deal with hostile aliens in a peaceful and nonviolent way. However, when her enemies do die, she rarely shows much emotion. The deaths of enemies are often followed immediately by humorous scenes. The most egregious examples are probably the the deaths of the Slitheen-Blathereen and the implied death of Miss Myers.
The Trickster is this for the villains, given that he's not just the main recurring foe (Slitheen aside), but his stories are considered among the strongest because of the emotional havoc he wreaks upon the cast - for many fans, the most disappointing thing about the series' cancellation (besides Elisabeth Sladen's death) is that the planned resolution to his storyline (which was meant to be the Season 5 finale) was never produced to give his character proper closure.
"Funny Aneurysm" Moment: In one episode, the characters have fun pretending to be sceptics when Rani's parents tell them that they just saw aliens. In a later episode, an alien takes over Rani's mother's body, and Rani even says that playing ignorant isn't very funny anymore.
Quite a lot of Maria's scenes with Sarah, and especially the way she talks about her in voice-over, suggest that she has confused adolescent feelings for her that may well stray into the erotic.
Idiot Plot: In "The Gift", why did the Blathereen give Sarah Jane the Rakweed? Why not just plant it on the sly on the other side of the world?
They wanted Sarah Jane to have a hand in the planet's downfall. For irony, or something.
What could be more ironic than one of the Earth's greatest defenders helping spread the cause of its ultimate destruction?
Periphery Demographic: The show, as we've said just above, has a massive one among classic Doctor Who fans. It's probably deliberate, as Sarah Jane is widely considered the best Classic companion in the show's history, and is almost certainly the most beloved.
It honestly veers into Multiple Demographic Appeal after a while. While, at its heart, it is a show intended for kids, there are so many references to Sarah Jane (and Jo Grant's) adventures with previousDoctors that the younger set just won't get. Add to that the rather attractive parents, storytelling that can stand with its parent shows, and Elisabeth Sladen saying at one point they even dabbled in Costume Porn (to apparently bring in the Carrie Bradshaw set), and it truly seems like they went out of their way to cover every base and make something everybody could watch together.
Replacement Scrappy: Some of the show's fans really didn't like Sky, and the fact that she was apparently meant to supplant Luke in the long-term just made things worse. However, the majority of fans were generally more forgiving, since the manner of the show's ending meant that Sky never got the character development she was supposed to.
Rani copped flak for this too, back when Season 2 aired and she came in as a replacement for Maria - the show not having much time to differentiate their personalities or give her much significance; her acting was also not considered the strongest even by critics. By the end of Season 3 she got enough Character Development and the actress had gotten comfortable in the role, so she was Rescued from the Scrappy Heap.
Rescued from the Scrappy Heap: The Slitheen in their second SJA appearance; removing the Toilet Humour and the necessity for their human disguises to be fat people goes a long way to making them more credible as villains.
Shipping: Clyde / Rani is the big one, but the fandom also ships Luke / Maria, Clyde / Maria, and occasionally Alan / Sarah Jane. On the homosexual side of fandom, Clyde / Luke and Sarah Jane / Maria are the most popular pairings.
Villain Decay: The Slitheen. After proving to be quite a threat in Doctor Who, similarly nasty (if a bit more farcical) in Revenge Of The Slitheen, and downright cruel in The Lost Boy they've been reduced to being largely comic relief, particularly in the Red Nose Day short From Raxacoricofallapatorius With Love.