Americans Hate Tingle: While the series is considered a cultural icon in its home country, rivaling successor series Dragon Ball, in much of the rest of the world, it's a cult hit at best. Part of this might have to do with the lack of international distribution, not helped by the fact that bringing it over to the west nowadays would mean having to break it out of the shadows of both Dragon Ball and modern anime, what with it being much more flippant and child-friendly than the typical image of Japanese animation. Ironically, it's likely that Dr. Slump would've had more luck cracking the western market had it been released there when anime was still under the thrall of the Animation Age Ghetto.
Crazy Awesome: It's Arale's hyperactive craziness that makes her so cool and popular as well as one of the most iconic Memetic Badass anime characters ever. That little girl can literally punch the world in half when she's just playing around!
Shortly after making his manga debut, Tsun Tsukutsun proved to be rather popular, especially with the girls. Though, having the ability to kick Arale's ass when provoked kind of helps.
Suppaman, as many fans remember him as one of the funniest characters. Other similarly pathetic characters like King Nikochan get this as well, to a lesser extent.
If there's something everyone remembers about this series, it's the sentient pink poop.
Fridge Brilliance: The "angel" twins are found from an egg left by "God" to halt the progression of technology after seeing the disasters it caused on other planets. This makes a lot of sense in retrospect when one reads Dragon Ball, where Goku and General Blue go to Penguin Village and Senbei expresses relief to be in a Shonen Jump again. It is later revealed in Dragon Ball that "God" and "The Devil" are two aspects of the same being and that "The Devil" can hatch "Demonic" offspring from eggs he produces. Later still, it is revealed several planets have their own "God", meaning the god who tried to stop the advancement of Earth was likely the same one later seen in Dragon Ball, which also explains the angel eggs (never mind the fact that God in Dr. Slump bears a closer resemblance to Master Roshi, though one could just consider this a case of Toriyama's typically heavy retconning). They were probably birthed by Earth's god the same way Earth's devil birthed demons.
While not iconic as Dragon Ball, "Arale" is still a popular and well-known series in Italy, and the anime, remake, and all the movies have been dubbed in Italian. In fact, many Italians love Toriyama for both "Arale" and Dragon Ball.
Likewise in Hong Kong and especially in Guangzhou, where the theme song was made into the unofficial anthem of the Cantonese language.
When Arale meets Donbe, a transforming fox, she claims that she can transform as well, and promptly spends the entire afternoon changing into an Ultraman costume. This almost feels like a retroactive parody of the Transformation Sequence scenes in Dragon Ball.
I Am Not Shazam: Nobody is named Dr. Slump. Senbei isn't even the main character, a fact lampshaded numerous times. Subverted in the unaired Harmony Gold English Dub where Senbei's name is changed to Dr. Slump. Also in the Italian Dub of the first anime, where he's only called Dr. Slump, and his real name is never mentioned.
Memetic Mutation: A popular reaction image of a bear in awed tears is from chapter 6 of the manga, when the bear that Arale and Senbei set free sees the mountains.
Moe: Arale in the first few chapters, she looks like a realistic 13 year old Meganekko and is extremely cute. She's still cute in her smaller childlike form (the one everyone remembers) but more in a "chubby little kid" way. Her innocence also helps.
The Scrappy: Tsun Tsuruten. He can best be described as a heavily Flanderized version of Senbei, but with none of the traits that made him an entertaining character. He is shown to perv on his wife and even his daughter, and can be a Toxic Friend Influence to Senbei when he tried to revert from his perverted behavior after marrying Midori.