The interaction between the Professor and Nano at the beginning of Episode 5, which plays out much like that between a mother and her child, has to be one of the cutest things in anime for a long time.
Episode 11: "Things we think are cool: A factory worker's shy smile"
Most of the "Small Miracles" and "Like-Love" segments.
Episode 13: The Professor is afraid of going to the bathroom by herself during a nasty storm, once she manages to get Nano to accompany her, the lightning scares the latter to her knees, huddling in fear, only for the Professor to comfort her, saying that it's not as scary when they're together.
Immediately followed by the Professor asking Nano if she wants to go to school and, at episode's end, getting her enrolled. Granted, the Professor was doing it partly to be able to snack more, but it's still heartwarming to watch Nano when she sees her uniform.
Episode 14: When Yuuko accidentally buys Mio some yakisaba instead of the yakisoba she wanted for lunch, the two end up getting into a huge argument. The argument itself is funny, but the end is very cute and heartwarming: in the end, they actually start praising each other, the same way they would insult each other. They finish their argument by raising their arms in the air as if they were going to punch each other... only to handshake and hug in reconciliation. Even Mai smiled at how cute it was!
The "Like Love" segment in Episode 15. Short, but very sweet.
So is the "Like Love" segment in Episode 16. It's a recurring thing.
One notable moment in particular happens in volume 6/Episode 23 when she visits the Shinonome Lab. Though her initial interaction with the Professor is a bit rocky, due to Mai unleashing her two dogs in front of the Professor and Sakamoto while they were taking a walk, they eventually bond when they start drawing sharks together. When Mai leaves, she pulls out one of the Professor's drawings, and you can clearly tell she's touched.
Episode 16: Yuuko gets proper confirmation that Nano's a robot, but pretty much says she doesn't care. Also, when Nano sees her lost screw left for her to find.
Yuuko having a time with the Professor, which is significant given the torment she faces on a regular basis.
A small but understated one from the second OP. Roughly 30 seconds in we see Mai standing behind Mio, fluttering the latter's pigtails about while Mio smiles adorably, swaying her head. Considering that these two barely speak a word to each other in-series, it's adorable to see them in this playful moment together.
One of the Professor's Character singles is "Hakase no Suki nano Nano", where she sings about how she hates Nano (because Nano is always getting mad at her, Nano makes her eat her vegetables, etc.) but also how she loves Nano. It's extremely cute and really pushes the Mother-Daughter / Big-Little sister theme the two have going on.
Her other single is "Hakase no Same to Inu", which is almost entirely her rapping about how awesome sharks are. It is adorable.
Professor and Nano's dancing segments.
Episode 25, period.
"It's unpredictable. That's why it's fun."
And, of course, the "Voucher for a Life-long Friendship", which should be filed under Tear Jerker as well. After all of the hardship and outright adversity Mio faced throughout the series, she gets a very heartfelt confirmation of friendship that is powerful in its simplicity.
Episode 26: Nano decides to keep her screw. Not that the Professor intended to actually remove it, anyway.
Professor: This little one spins when you're really happy!
And then, in the midst of all the chaos, when cakes are ruined and everyone's panicking, the screw is still spinning.
One that doesn't immediately draw attention to itself in the same episode: Yuuko accidentally pulls off Nano's arm and her immediate reaction is to worry that it hurt her.
Yuuko's clumsiness makes life better for everyone.
A small one, but the last Go-Soccer segment. Even if Sekiguchi Cannot Spit It Out, there is a sense that Daiku at least somewhat understands what she wants to say, and responds in a simple, yet sweet way.
A similar example, when Sakurai asks Mio what she wants to do, which she sheepishly writes on the forum, "mangaka". Mio is clearly embarrassed by this, probably not helping what kind of art she usually draws, but Sakurai complements her that she is a great artist and should follow her dreams.