She's enjoying that a bit too much.

With the show's large cast of female characters, this is pretty much unavoidable.

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Les Yay

Batgirl/Wonder Woman

The closing shot of the theme song, which shows a yearbook photo of the girls, shows both of them hugging each other in locked arms. They're the only ones in said picture to touch each other.
Wonder Woman gives a Rousing Speech to the girls while clasping Batgirl beside her. Batgirl virtually melts in her arms with an adorkable, blissed-out grin.
- Not to mention when Wonder Woman and Supergirl are brawling before that, Batgirl actually gets hearts in her eyes from watching their superpowered catfight.
After things go wrong during the battle at Sweet Justice, they're the only ones who still want to be heroes. Plus, when Diana is being sent back home, Babs is the most concerned for her and is the most focused on trying to get her back.
The ending to "#Frenemies" has them hold hands with each other and apologize.
Barbara actually dresses up as Wonder Woman for Halloween.

Batgirl/Harley Quinn

Harley throws Batgirl a kiss after they finish talking in the pilot.
Before running off to suit up as her villainous alter ego to get revenge on Robin for humiliating her best friend, Harley kisses Batgirl on the head in "#GothamCon."
Harleen and Babs act a lot like Sickening Sweethearts in "#Frenemies". Babs is so close to Harleen that she wants to die on the same day as her, so that they can have a best friend-themed funeral.
Harley and Barbara wearing matching Wonder Woman Halloween costumes, and hearts floating off their heads as they see they're wearing the same costume and hug.
- And later in the episode:
  Barbara: Even after all this, after everything, it's still me! Babsy! And I love you...
  Harley: ... I love ya too, Babs...

Batgirl/Supergirl

When Kara rescues Babs after Diana had thrown her into the air, Babs has a blissed-out look on her face while Kara gives her a Bridal Carry.
Babs can be seen at the movies with Kara at the end of "#SilverScreamer".
Kara gives Babs a hug at the end of "#SuckerPunch".

Zatanna/Supergirl

"AdventuresInBunnySitting" has two notable instances of this. One is at the beginning, where after she leaves to go to the birthday and lets her watch the bunnies, Zatanna shakes her hips at Supergirl and says goodbye in a rather flamboyant, over-the-top manner. The second time is near the end, when the bunnies get out of hand—their argument is like that of an old married couple.
The short "#WasabiWar" essentially plays out like a date between the two of them.
Kara dresses up as Zatanna for Halloween.

Green Lantern/Poison Ivy

In "#MisgivingTree, Jessica's really determined to become friends with Pam Isley, to the point of following her everywhere she goes. Then her friends' intervention to discourage Jess, especially Zee's emotional outburst, are like a support group after failed relationships/marriages.
- We even have Wonder Woman gently trying to break the idea that Jess's "amicable advances" aren't working. Zatanna takes the more direct route and screams "She's just not that into you!"
On the other side of things, we have "#Vegecide", where Pam sees Jessica drinking a veggie smoothie, and is absolutely shocked, almost like someone who just caught their significant other cheating on them—she then sends out her plants, not unlike a clingy, jealous girlfriend.
In "#ScrambledEggs" Jessica grabs Pam in a hug and grins when she finds out they're assigned to work on a class project together.

Zatanna/Bumblebee

Zee and Karen act almost in a flirtatious manner in "#IllusionsOfGrandeur". In particular, Zee is very open to confessing her feelings to Karen, and she sees her as a worthy assistant. Karen is also willing to help set Zee straight when she's letting her ego get the better of her. At the end, when Karen gives her recognition when no one else does, she's quite touched.
#AbracadabraPalooza" has Karen getting a zit, and Zee telling her a story of how she also dealt with self confidence issues with her body. She's shown taking this quite personally in trying to help her.
#DramaQueen" has Karen once again being Zee's assistant in making sure the play turns out well. Both are also very in sync with each other.
#LivingTheNightmare" has Zee going into everyone's dreams when they have nightmares. While Zee's concerned for all of her friends, she seems the most concerned for Karen (the first of whose dreams that Zee enters).

General/Other

After Diana sticks the landing on her vault and is mobbed by her cheering teammates, one of them (the dark-skinned redhead second from the left◊) is staring at her with a look of starry-eyed awe that suggests she's much more than professionally impressed with Diana.
Played for a quick Bait-and-Switch joke in "#SoulSisters", when Green Lantern yells, "Stop that!" For a split second, it looks like Supergirl is kissing the unconscious Catwoman, but is actually Face Doodling her.
Tatsu and Diana have a lot of chemistry and quickly become close friends due to their shared interests.
Batgirl's dinner with Rose and her dad feels a lot like a "meet the parents" date taken out of context.
Batgirl, in general, has a lot of chemistry with a lot of women.
"#HousePest" reveals that Jessica's the daughter of a lesbian couple.
"#TheFreshPrincessOfRenFaire" has Zee and Diana arguing like an old married couple, and later Zee becomes the Fair Maiden with stars in her eyes and Diana her Knight in Shining Armor, and then they're back to arguing.
Jessica gets embraced and cuddled by Kara under hypnosis in "#AngerManagement".
Jessica: (giggling) That tickles!
For Halloween, Karen dresses up as Green Lantern, Jessica dresses up as Batgirl, Diana dresses up as Bumblebee, Kara dresses up as Zatanna, and Zee dresses up as Wonder Woman like Harley and Babs.

Ho Yay

The Invinci-Bros

Hal is really determined to be friends with Superman.
In the episode "Drama Queen", although Oliver Queen and Mortimer Drake were competing with each other for the lead male role in the play at first, after their fight as Green Arrow and Cavalier, Oliver is impressed by Mortimer's dramatic death scene and Mortimer is impressed by Oliver's performance and the two boys come to respect each other while criticizing Zee's acting and direction.
Sinestro's entire motivation for turning evil was to get back the attention of Hal.