Accidental Hero: Believe it or not, there are actually a number of times when Gadget is legitimately helpful and actually contributes towards solving the case. It's just that he tends to do it purely by accident.
Repeated line whenever Gadget is congratulated for solving the case: I did?
Adults Are Useless: However, when Gadget knows Penny is in danger, he becomes scarily competent.
Fearless Infant: Gadget was shown to be one in a flashback of the final episode, where he is seen as a baby witnessing crime boss Spuds Malone gun down people with his potato gun the Red Rose and being completely unfazed by the chaos.
Horrible Judge of Character: Gadget always believes the lies MAD agents tell him. He'll even trust MAD agents that aren't even bothering with deceiving him and are just trying to kill him ("Ah! The welcoming commitee!"). This goes all the way to the opposite end of the spectrum. He'll spot Brain in disguise, and instead of recognizing Brain, he'll think Brain is the MAD agent!
Let's Get Dangerous: Very, very rarely, Gadget showed competence and was able to avert disaster all on his own. This tends to happen when Penny is in danger.
Man Child: He occasionally acts this way, such as in "Gone Went the Wind", when he becomes snowblind, mistakes a polar bear for Santa Claus, and sits on the polar bear's lap asking for a choo-choo train, a candy cane, and a doll of himself.
Papa Wolf: When he knows Penny is in danger, Gadget becomes virtually unrecognizable (and unstoppable) as he becomes the superhero he is supposed to be. For what it's worth, one of his few talents seem to be as a guardian, when not oblivious to some villain on her tail, he raises Penny rather well and gives rather intelligent life lessons.
This also extends to other people Gadget cares about. He's gone to similar lengths to save Brain (when he actually knows that he's dealing with Brain, as opposed to thinking the disguised Brain is a MAD agent) and Professor Von Slickenstein, the scientist who gave him his gadgets.
Super Strength: While usually not demonstrated in very ostentatious fashion, Gadget's baseline strength is clearly superhuman. There are occasional exceptions where it's more obvious; for example, in "Gadget's Gadgets" he broke free when literally riveted in place with steel bands on an operating table.
Super Toughness: Gadget frequently suffers more extreme abuse than RoboCop has to put up with on a bad day. He has survived falls off skyscrapers, crashing airplanes, electrocutions, any number of bombs and IEDs, and even being sawn in two. As yet, nothing has permanently crippled him.
Corrupt Corporate Executive: For the movie. Also any episode that revolved around MAD setting up a legitimate business and then wiping out their competition by some underhanded means, leaving Dr. Claw with a monopoly.
Enfante Terrible: Was implied to be one when he was a child. He mentions that he desired possession of the Crown Jewels since his youth in "Gadget's Clean Sweep", his mentor Les Renowned introduced in "Gadget meets the Clan" can be seen in a picture lecturing a younger Claw holding a lollipop, and the 1992 Christmas special had Dr. Claw mad at Santa Claus for never getting him the dirty tricks sled he wanted as a child.
Exact Words: From the episode "The Invasion": "I said I feel like singing!"
The Faceless: Not only faceless, but also bodiless as only his hands and arms are ever shown to the viewers. Luckily, in the movie, his whole appearance is RupertEverett.
His action figure however depicted his face. Click at risk of shattering childhood memories◊. In a brilliantly devious ploy of marketing, the packaging for the figure covered up his face with a piece of cardboard inviting you to buy the toy and finally see what he looked like.
Joker Immunity: As shown in the second live-action film and the animated direct-to-video film Inspector Gadget's Biggest Caper Ever, not even capturing him and putting him in jail can stop Claw.
Karma Houdini: Usually played straight in that he always avoided getting arrested, but some episodes subvert it and have Claw suffer a minor problem as he makes his getaway, such as being trapped at the bottom of the arctic waters in "Gone Went the Wind", getting frozen in "The Quimby Exchange", and getting hit by the blast of the Red Rose potato gun in the final episode "Gadget and the Red Rose".
Nice Girl/Friend to All Living Things: Penny rarely shows anger, hardly ever even gets annoyed (even with her uncle's antics), and the number of people she shows intense dislike for throughout the entire franchise can be easily counted on one hand. As a result, she has a knack for making friends and allies at the drop of a hat.
Non-Standard Character Design: In comparison to the other very cartoony designs in the show, Penny's is noticably more anime inspired and realistically proportioned. Gadget And The Gadgetinis gives her a somewhat wackier design, though still slightly more subtle than the rest of the cast.
Not so Above It All: She may be the most competent and mature person in her household, but mention her favorite singer and she still goes into shrieking fangirl mode.
Only One Name: Penny was never given a last name, though a lot of people just assume it's Gadget.
Parental Abandonment: It's confirmed in the second season that Penny's parents are apparently deceased. We only have Dr. Claw's word for it but according to him, Penny is Gadget's only living relative.
Action Pet: Although more wacky than outright action, he still goes through things that one would have to be tough to get though.
Beleaguered Assistant: Occasionally averted when Gadget actually saves Brain. This usually happens when Gadget thinks he's a MAD agent and is trying to capture him, and ends up grabbing Brain right before he'd fall to his death.
My Instincts Are Showing: Though usually a neurotic Straight Man with more human intelligence than most of the police force combined, Brain sometimes has displays of puppy-like friskiness. For example, at one point when Gadget tries to throw away some explosives, he mistakes it for a stick and excitedly plays fetch with it.
Chew Toy: Without fail his 'self-destructing message' will be near him when it goes off.
Composite Character: Of the Chief of CONTROL and Agent 13 from Get Smart, which Inspector Gadget could be considered a Spiritual Successor to. Like the former, Quimby gives orders and mission debriefings to the bumbling protagonist. Like the latter, he has a tendency to hide in different improbable locations while giving messages to Gadget.