A few moments are quite scary, most of them involving Anthy and Akio, most of which aren't "jump out of your seat and scream" scary but a more psychologically disturbing kind of scary, especially once you find out the true nature of their relationship.
The first real example of nightmare fuel came with the first episode of Black Rose arc, showing us how Mikage and Mamiya recruit the Black Rose Duelists. Akio's fiancee Kanae goes to Nemuro Memorial Hall for her interview. She enters an elevator and, in anguished tears, divulges her attempts to make friends with Anthy as the elevator speeds past strange plaques (actually mausoleum plots). Meanwhile, someone over a loudspeaker tells her to go "Deeper. Even deeper" as the elevator accelerates more and more as it descends. Finally, poor Kanae reaches her breaking point and Mikage appears behind her from the door, telling her she has no choice but to revolutionize the world. And then before she can react, Mamiya stabs her in the chest with a black rose. Poor Kanae!
To add some extra Fridge Horror to that, the Mamiya we see throughout most of this arc, including this scene, is an illusion acted out by none other than Anthy; in other words, Anthy is the one who stabbed Kanae! Brr.
Then there's the apparent poisoning of Kanae by Akio and Anthy. Remember that she's someone we don't see at the end. Definitely didn't get a happy ending of any kind.
Not to mention you can hear the 100 dead boys kicking around inside their coffins.
Some of the later shadow girl plays sneak into this territory. There's even a borderline Gory Discretion Shot when one of the shadow girls breaks the other one in half. A disturbingly realistic squishy, snapping sound is included, of course, and somehow, B-ko's deadpan "ow" afterwards makes it even worse.
The one in episode 34 has as subject Akio and Anthy's story. They also interacted directly with Utena and invited her to their theater to make sure she watched it... And she came with Akio and Anthy.
Episode 9: Touga and Saionji find a young Utena hiding in a coffin next to her dead parents (who are in separate coffins). The trauma of losing her parents has caused her to give up on life, as she thinks it's pointless to live on if all humans eventually die anyway. She's no older than six or seven in this flashback, and yet she's basically suicidal.
Later in the same episode, Anthy appears similarly curled up in a coffin. Made more chilling in that the whole point of her doing so was to remind Saionji of his childhood trauma and ultimately bring about his downfall.
Nanami got a kitten for her brother as a birthday present, but being such a Yandere for him she drowned it because it was taking up all Touga's attention. She was seven years old at the time!
The relationship between Nanami and Touga has a lot of this, thanks to her sheer obsession with him and the way he emotionally manipulates her. It all reaches a head in episode 32, in which Touga suddenly tries to rape Nanami in Akio's car (which she fights off in horror) and gets angry because if sex isn't what she wanted, then what does she want? Actually, the sheer level of bullshit he pulls on her in that episode is nightmarish in itself.
This careens straight into fridge horror territory if you buy the theory that he shares the same fate as Touga in the movie, and was molested by his step father That Touga's idea of family love has been so completely warped makes the entire mess between the two of them chilling, to say the least.
The scene in episode 3 when Nanami tries to embarrass Anthy by making her dress dissolve. Also, there's also the Uncanny Valley view of the crowd at the party from Anthy's point of view. It doesn't seem so horrific at first, but when you rewatch it with the knowledge of why Anthy doesn't like crowds, it's decidedly painful.
There's also Kozue who is shown to be rather obsessed with her twin Miki and to grab her brother's attention she exercises loose sexual morals. She's extremely possessive of Miki to the point where she physically hurts anyone who tries to hurt or "steal" him.
In one episode, Miki's piano teacher is hinted to have a less than savoury interest in him, and he's inappropriately touchy-feely with him when they talk. Kozue has clearly noticed; it's later revealed that she pushed the piano teacher down the stairs in order to protect Miki from the piano teacher's predatory intentions.
In Episode 8, Utena and Anthy switch bodies. After Nanami spends a bunch of time trying to find the super-rare chili powder so they can recreate the curry that caused the body switch, it's discovered that Anthy didn't actually use the super-rare chili powder and that the curry she recently made has the same properties. Saionji and Chuu-Chuu then switch bodies using the new curry, which implies that Saionji spent at least some time in the body of a tiny monkey. If that isn't traumatizing, I don't know what is.
On that note: Utena and Anthy spend AT LEAST a week with switched bodies. Akio did NOT keep it in his pants long enough for Anthy and Utena to switch back, which means that Akio probably saw Utena naked long before he and Utena were actually acquainted.