Barbara Gordon

Earth-One / Post-Crisis / New Earth

 * 1)  Dick Grayson / Robin / Nightwing -  Dick had strong feelings for Barbara when they were members of the Batman family, but she did not appear to reciprocate them at first. She was a few years older older (25 when he was in college, although this may have been retconned by the Post-Infinite-Crisis Nightwing Annual #2, which mentions them both being 16 at the same time).

Dick confessed his feelings to her after one mission, before he realized she appeared to be asleep. She was, however, only faking, and she was beginning to reciprocate. But there was much going on in Dick’s life, however, including him rebelling against Batman and his life, and Barbara didn’t know where that would leave her.

Some time later, she went to his apartment to tell him how she felt, but the door was answered by Koriand’r in her underwear, prompting Barbara to leave upset.

The two later entered a relationship when Dick moved to Gotham's neighboring city Bludhaven. There were engaged at one point, but held off after Dick was nearly killed in Infinite Crisis.

2.  Jason Bard  - Barbara began dating private investigator Jason Bard after that, and the two eventually became engaged.

After Barbara was paralyzed, she ended her engagement to Jason. She and Dick later slept together, after which he gave her an invitation to his and Kori’s wedding, causing an infuriated Barbara to kick him out. This type of behavior was later attributed to an evil possession on Dick’s part.

After Dick and Kori broke up, and after Barbara became Oracle, she and Dick began a relationship, eventually becoming engaged at the time of Infinite Crisis. However, one year later, they had broken up.

Just before Identity Crisis, it is also shown that Barbara is the unrequiting recipient of the affections of Ted Kord / Blue Beetle.

New 52

 * 1) Dick Grayson / Robin / Nightwing - These two shared a working relationship and an apparent attraction.
 * 2)  Luke Fox / Batwing  and Barbara also shared an attraction.

Dick eventually made it clear he wanted more with Barbara. Barbara, however, chose to date Luke instead.

Adaptations
 DC Animated Universe 
 * 1)  Dick Grayson / Robin / Nightwing  - Dick and Barbara romance each other even before they realize each other’s alter egos. They become a couple for a time before breaking up.
 * 2)  Bruce Wayne / Batman  and Barbara had a brief relationship, according to Batman Beyond. In the comic books continuing the DCAU, she had a pregnancy that resulted in a miscarriage after she had ended the relaitonship.
 * 3)  Sam Young  was the man Barbara eventually married before becoming police commissioner.

 Birds of Prey 
 * 1)  Bob , a teacher, becomes Barbara’s main love interest. They date, him never knowing her to be a coordinator of vigilantes. Things hit a snag when Bob’s mother tells her she doesn’t want her son dating a woman in a wheelchair (shots fired!) but the relationship continues. Bob is killed in the finale by Harley Quinn.

Arkham Series
 Batman: The Killing Joke (movie) 
 * 1)  Tim Drake / Robin  and Barbara apparently experience romantic tension in this series, even apparently being married (in the future?)
 * 1) Bruce Wayne / Batman - Batgirl lusted after Batman in this controversial adaptation. This culminates during an argument, which ends in them kissing, in costume, before shedding the costumes and having sex on a rooftop.