The dance scene certainly qualifies, especially after everything the family has been through by that point. Everything taught them to stand by one another, and take risks even when it seems scary and pointless.
Also when Dwayne is having his breakdown and refuses to get back in the van. Nothing anyone can say has made any difference, and Richard suggests that Olive try. Without saying a word, Olive goes down the hill and puts her arm around him and lays her head on his shoulder. Such a perfect testament to the power of just being there for someone, even if you can't make everything okay. And it gets even more heartwarming right after, as Dwayne, after having yelled at his mother, accusing his whole family of being losers and screaming how much he hates them all, reacts to Olive's hug with no more fuzz a simple "Ok. Let's go," and then easily gets up and helps his little sister back up the hill, so they can get back on the road and get to the pageant on time.
Earlier in the movie, when the familly leaves a gas station after an arguement, they forget Olive. Dwayne is the one who points it out, getting them to go back. It's not much, but it's still very touching to see that he cares for his sister even though he claims to hate everyone.
The night before the competition, Olive is starting to feel the pressure, feeling insecure about her looks and actually starts crying. Grandpa swoops in, after being an asshole for 90% of his screentime, and tells her not only is she beautiful, but she's the most beautiful girl in the world. Seeing how Olive is coming apart just the way a normal little girl would and how Grandpa is comforting her was really touching.
The very subtle bonding Frank and Dwayne have as the movie goes by, culmunating in their first actual talk on the bridge.
Olive: Do you eat ice cream?. Miss California: Yes. My favorite is Cherry Chocolate Garcia, except technically I think it's a frozen yogurt.
And the whole thing is just so sincere and so perfect, it feels like Miss California knows exactly why Olive is asking the question.
Not to mention that the same judge is the only one who loves Olive's performance.
After Richard's book deal falls through, his father (who throughout the entire movie has pretty much bad-mouthed Richard and treated him with disdain), tells him how proud he is of Richard for being brave enough to take a chance.
All of the family members, even Dwayne, tempting Olive into eating ice cream.