Sophia Di Martino, the actress for Sylvie, has spoken in an interview about why Sylvie only has one horn at the start. It's because she's been on the run a long time doing a lot of fighting. It is also meant to represent her character at the point we meet her though, she's kind of broken and lost in the world and so is her crown.
Sylvie's been on the run for a long time, she's been hiding out in apocalypses. She's been brawling a lot, she's been beating a lot of people up. I think the horn is probably broken somewhere along the line. It's also, I believe, inspired by the comics and the Lady Loki character. It says a lot about Sylvie - she's wearing a crown, but it's kinda broken.
ScreenRant, Sylvie's Broken Loki Crown Explained By Actor
Sylvie eventually loses her crown on the train during the fight in episode 3. Again this is meant to help show the growth of her character. She doesn't need to hold onto the past anymore or hopes of ruling (we get more hints at this in episode 5) and she's growing as a person. We see her come more into her own and in a way move past the "Loki anger" and become Sylvie even more. She's also ready to leave the shadows behind and face the world.
What was it like opening up those fissures in the Loki pompousness and Loki attitude that you see in The Avengers in moments like the train scene?
The train scene I love because Loki doesn’t get many wins and it’s nice to see him having a nice sing-song. He’s just enjoying himself. Because I think that’s such a funny way, as well, to show the difference between him and Sylvie is that she’s on a mission. She’s like, “We’re going to get off this moon.” And when she’s offered a drink, she’s like, “No, thank you.” But Loki’s like, “Yeah, I’ll take the drink.” I remember I had this diagram with Christine from costume. People will see it as the show goes on, but across this episode in particular, Loki’s lost his Variant jacket. He doesn’t have that by the end, and it’s like, as a Loki, he’s transforming and in that sense, it’s almost like his outer appearance does as well, in the sense that his jacket’s gone, but also he gets more dirty. And Sylvie’s the same. She loses her horns on the train, and she loses that coat that we’ve seen her hiding under in the first two episodes.
thrillist, How 'Loki' Director Kate Herron Built Loki and Sylvie's Bond
For a more general rule on this though, which you might want to apply to other Lokis and you might not, Hiddleston told Kate Herron, the director, the below.
Herron was a big fan of the God of Mischief prior to signing on to direct the six-episode first season of "Loki," but she learned new things from Hiddleston.
One of her major takeaways was Hiddleston's interpretation of Loki's signature gold horns.
"He always felt that Loki wears the horns when he's feeling his most powerful or most in control, and I just thought that was really interesting," she said.
Insider, 'Loki' director Kate Herron says Tom Hiddleston's 'Loki lectures' on the Marvel character 'brought everyone together'
You can likely read into that regarding the other Lokis we see. Though it doesn't appear to be universally true when we consider that Boastful Loki never wears horns/a crown.