Base-Breaking Character: Ruby. More specifically, her alternate personalities. Some find them amusing and add variety to the story while others seems them as too far removed from the original and annoying.
Cargo Ship: Yang becomes very attached to the vehicles in the canyon, alternatively sympathizing with and becoming jealous of Caboose and his love for Sheila. The Pelican caused her to faint from sheer joy, and the mere sight of Freckles on the DVD cover caused her to have to...um, use the bathroom for a while.
Creator's Pet: Noire began flirting with this in the Season Five viewing. She effectively replaced Blake and garnered sympathy with the others via a fake back story. The increased prominence got her some hatred, which the author got snippy at people for wanting her gone right away. Some in-universe comeuppance for her actions and the promise of her going back to her original role by Season Six helped to keep her from going too far.
She came back as her own seperate character in Season Seven.
After Church is revealed to be the Alpha AI, Weiss has an emotional breakdown, thinking that she isn't human and was built by the Schnee family to be a perfect heiress, which didn't work out. After Ruby punches her in the stomach, Ruby goes on an incredibly sweet spiel about how Weiss is really human, and even if she wasn't, it wouldn't change anything.
Ruby: You're my best friend, Weiss, and I won't stay sitting down while you tell yourself that you're a mistake. […] Ruby: The point is that you are Weiss, not an A.I or W-3155 or some shit. And even if it turns out you really are one, nothing would change. We're here for you.
Noire goes through an existential crisis and laments that her entire life is nothing more than a lie and that all she is capable of is lying further. Blake, who has more than enough reason to hate Noire, conforts her and tells her she is more than that.
In-universe example - aside from his name, the girls have nothing but respect for Captain Flowers.
And they seem to like York, which is saying quite a bit.
Out of universe, Noire is fairly popular for her fourth-wall breaking antics and pervy jokes, though her increased prominence for the Season Five viewing did lead to some backlash from overexposure.
Les Yay: Obviously common with the RWBY team, given the preferred ships of the fandom. At one point Ruby makes a bet with Weiss and the loser has to kiss the other on the lips. They both accept, then promptly freak out when they realized what they agreed to. Meanwhile, Blake had to...um, use the bathroom for a while.
Narm: Some of Creepy Ruby's attempts to be scary probably won't all come off as creepy to the readers as it does to the girls.
Nausea Fuel: In-universe, Donut discussing his health care ritual.
If you think about it too much, Yang being horrified to the point of wetting herself for well over an hour.
Ron the Death Eater: In-universe, Ruby has a very exaggerated take on Tex because of her rocky romance with Church and really wants to see her stay dead. This makes her truly horrified reaction to Tex's Heroic Sacrifice into a genuine Tear Jerker.
Tear Jerker: Noire gets some due to her Meta knowledge. She is fully aware that she is present in a fanfic, and when her guard is down, laments that she believes it means that her entire existence is a lie.
She also actively dreads the story ending, because she realizes that she will be as good as dead when it does.
Took the Bad Film Seriously: Not that Red vs. Blue is a bad show, but team RWBY reacts far more strongly to the series than the vast majority of regular watchers, seemingly forgetting the comedic nature of the story.