Kara Zor-El / Supergirl

Earth-One
Growing somewhat unstable, Brainiac 5 created a robot duplicate of her in his sleep, convincing himself this was the real Supergirl (Superboy #204). At one point, he asked her, “Please stay, and be my girl,” but she could not. I’m not sure what issue that was, or what anything else is, but let me know if you know.
 * 1)  Brainiac 5  and Supergirl formed a mutual attraction as she sometimes joined the Legion of Superheroes.

It was also revealed that Superman actually held romantic feelings for his cousin (Action Comisc #289, 1962)

2.  Biron / Comet / "Bronco Bill"  This individual was originally an ancient Greek centaur named Biron. The witch Circe gave him a potion to turn him fully human, but by the machinations of the same enemy, she mistakenly made him fully horse instead. To make up for this, Circe gave him super-strength, telepathy and immortality. His enemy later imprisoned him in an asteroid.. However, when Supergirl's rocket passed, it broke the force field, enabling him to escape. He beame Supegirl's pet horse, dubbed Comet the superhorse, with her totally unknowing he was a sentient being.

Later, the two went on a mission with her to the planet Zerox, where a magic spell was cast that turned him into a human, but only while a comet passes through the solar system he is in. As a human, he adopted the identity of "Bronco" Bill Starr, a rodeo trick-rider, He would occasinoally romance Supergirl, who was totally unaware he was actually her pet horse!

3. Jimmy Olsen - Apparently these two (in her identity as Linda Lee) had an attraction, and even dated on-and-off for a time.

4. Dick Malverne was the main attraction Linda Lee harbored, and her main love interest through much of the Silver Age.

--The New Earth Supergirl also has an unrequietd romantic/sexual attraction to Nightwing

New 52 / Rebirth

 * 1)  Michael  (after the Doomed storyline)
 * 2)  Ben  (Rebirth)
 * 3)  H'el  - All I know about this H'el guy is he is an enemy of Superman, so Supergirl falling for and kissing him, which she does, is essentially forbidden love.

Imaginary Tale

 * 1)  Jimmy Olsen -  In an imaginary tale in an issue of Superman's Pal, Jimmy Olsen, Jimmy and Kara, in her identity as Linda Lee, got married. She only later revealed her secret identity to her husband.

Supergirl (movie)

 * 1)  Ethan  - In this movie, Argo City has not been destroyed, and Kara comes to Earth looking for an object to save it. While undercover as Linda Lee, a student at an all-girls boarding school, she falls for the gardener, Ethan, who reciprocates. The evil witch Selena who she is fighting also finds herself enamored with Ethan, and taken captive. Sueprgirl rescues Ethan from the Selena's clutches. However, she ultimately leaves, going back to Argo City.

DC Animated Universe

 * 1)  Brainiac 5  - After travelling to the future and teaming up with the Legion of Superheroes, Kara falls for Brainiac 5, and decides to remain with him in the future.

Smallville

 * 1)  Jimmy Olsen  - Kara is attracted to Jimmy Olsen, an attractoin that is briefly entertained while he is split from Chloe Sullivan. She kisses him at least once. But the bond dies, and he gets back together with Chloe.

Arrowverse
James, meanwhile, begins to also fall for Kara, in spite of being in a relationship with Lucy Lane. Eventually, Lucy catches on to his bond with Kara (only knowing her by that name in her civilian guise), and breaks up with James. James and Kara share one kiss in the first season finale, but, bizaarely, she then tells him at the beginning of the next season that she knows they work better as friends.
 * 1)  James Olsen  - After meeting James Olsen, who no longer likes “Jimmy,” she immediately falls for him. Knowing her identity, he becomes a member of her unofficial team, along with her friend Winn Schott Jr. Winn is attracted to her, and he ultimately makes this known, but she does not reciprocate, and he moves on.
 * 2)  Adam Foster  - Kara helps Cat Grant get back in touch with her son Adam. Kara and Adam then begin dating, but her duties as Supergirl make her feel the relationship is doomed, and she breaks it off.
 * 1)  Mon-El , at this time, comes to Earth, and is sheltered by Supergirl and her friends. Although he as at first an immature cur that she can’t stand, he grows up a lot in a year, and the two become a very serious romantic couple. His evil mother then invades the Earth with an army of Daxomites and, to ward them off, Kara is forced to fill the Earth’s atmosphere with small amounts of lead, their weakness. Mon-El is forced to leave the planet. He later returns, having traveled to the future and gotten a cure for the lead-weakness. However, for him, it has been seven years. He is only in this time on a mission for the Legion of Superheroes, and is now married to Imra Ardeen.