Scarlett is definitely not a doting mother, especially to her two oldest children. Yet when the Yankees raid Tara and Scarlett keeps her mouth shut - until they try to take Wade's sword. Scarlett's genuine reaction, that she can handle seeing them take anything but her little boy's most cherished possession, is both heartwarming and tear-jerking.
Any of the moments when we see that Scarlett truly does care for her children.
One of the major things that drives her to make more and more money after the war is to make sure Wade will be safe and provided for, that he will be able to attend a good school and that he will have a good inheritance. She completely forgets the social status angle but it's not because she doesn't love her boy, it's because it doesn't occur to her that such a thing would matter anymore.
Rhett comforting Scarlett and promising he will help her get home to Tara, the evening of the first burning of Atlanta.
Scarlett's reaction when Melanie miscarries and they know she's not going to make it. It's both heartwarming and heartbreaking.
Likewise when Scarlett and Ashley are caught embracing and Scarlett's biggest fear is what Melanie will think.
The talk Rhett and Wade have on the day Bonnie is born.
When Pork gets shot trying to steal food for everyone at Tara Scarlett tells him how grateful she is to him and promises to buy him a gold watch when she can afford it. When Gerald dies she gives Pork his watch, knowing how much it would mean.
After the rumours that Scarlett and Ashley are having an affair start, Rhett forces Scarlett to attend Ashley's party by herself in a provocative dress. Melanie's reaction to Scarlett's entrance is to walk over, kiss her on the cheek while complimenting her dress, lead her over to Ashley, and tell him to get Scarlett some punch. This also arguably qualifies as a Crowning Moment of Awesome on her part for making it absolutely clear in her usual calm, polite way that her trust in her husband and sister-in-law will not be broken by gossip.
When Melanie personally thanks Hooker with a Heart of Gold Belle Watling for providing an alibi for Rhett, Ashley and Dr. Meade after the shantytown raid. Though Belle is concerned about what this might do to Melanie's reputation, Melly dismisses such talk and says she's proud to know Belle.
Melanie giving up her wedding ring for the Confederate cause, and a genuinely-touched Rhett redeeming and returning the ring to her.
During a raid at Tara Melanie first flees with the others but once she's made sure they are safe she returns, even though she's still terribly weak from giving birth, and is the only one to show up and help Scarlett save Tara (also another Crowning Moment of Awesome for both ladies). Afterward even Scarlett has to admit that her sister-in-law is always there when you need her.
After Scarlett's miscarriage she does call for Rhett, even though he doesn't know it. Mostly though she calls for Melanie, needing her more than anybody else. It's actually debatable whether Rhett Butler is the great love of her life or if Melanie Wilkes is, albeit in a platonic sense of the word.
Scarlett is actually very supportive and helpful to Melanie during Beau's birth, including encouraging her to forgo customs and scream as much as she likes. In the movie she's very gentle and understanding when Rhett asks what Melanie wants and she knows it's Ashley's picture and Charles' sword - things a pragmatic like Scarlett would probably feel are completely unimportant at a time like that.
YMMV because of the paternalistic racism, but Charles told Scarlett about how much Uncle Peter shaped his and Melanie's lives after their parents died and were left with a ditzy and shrill Aunt and a He-Man Woman Hater Uncle who berates said Aunt. Peter raised them, made the decision that Charles will attend Harvard instead of just the schools in Atlanta and when Melanie will put her hair up and attend balls.
Early in the film, Ellen O'Hara comes home. While Suellen and Careen immediately clamor for her attention, the first person Ellen turns to is Scarlett and asks if she's all right, saying she's been worried about her. Despite Scarlett feeling later on in the film like she's been nothing but a disappointment to her mother, Ellen does care for Scarlett and wants what's best for her.