This is established right at the beginning where Basil is able to figure out Dawson's profession and where he came from in less then 3 lines of shared dialogue!
Badass: Say whatever you want about Ratigan, but taming a cat and making it into your minion is pretty cool. Especially if you're a rat... I meant, mouse. Same goes for Basil who managed to befriend a dog.
When Basil tries to press his button by calling him a rat, Ratigan freezes, spawns Technically a Smile, informs him that he would have stuck around to watch his death but Basil was late, and walks away cheerfully.
Big "NO!": Basil, when discovering that the two bullets don't match.
Big "Shut Up!": Ratigan, after Olivia starts to bad mouth him when he's escaping.
Bizarre and Improbable Ballistics: The idea behind the ballistics test is presented fairly well, but the experiment conducted to determine a match has a fatal flaw. Basil holds the ends of the bullets together, and most of the grooves etched into the bullets by the rifling match, except for few. The problem is, if you turn the bullets around in your head so they're side-by-side as opposed to end-to-end, one would be a mirror image of the other! Thus, a cursory glance at them would be all you need to see that they don't match. If Ratigan knew Basil would make such a stupid mistake when comparing the bullets, this could be a case of Fridge Brilliance, as it implies almost impossibly good foresight.
Bond Villain Stupidity: Justified. Ratigan does build a giant death trap that he leaves our heroes in. However, it is explained that he wanted to stay and watch it, but as the heroes arrived 15 minutes later than he planned, he doesn't have the time to stay. So he rigs a camera to capture Basil's dying moment instead.
Brick Joke: Right before leaving for Buckingham Palace, Ratigan tells a soon-to-be-dead Basil "Now, you will remember to smile for the camera, won't you?" as he activates his deadly Rube Goldberg contraption which is set to take a picture at the exact moment Basil and Dawson are to die.
...A few scenes later, after Basil's last-minute ingenuity very narrowly lets them escape the trap, he grabs hold of Dawson, catches Olivia and says "Smile, everyone!" as the picture of the three is successfully taken.
Bunny-Ears Lawyer: Basil is an undeniable genius who, when we first meet him, comes off as quite possibly insane.
Butt Monkey: Ratigan has a tendency to treat Fidget this way.
The Cameo: Sherlock Holmes himself appears three times in silhouette. In the books, Basil actually lives under Holmes' floor. From his and Watson's dialogue, Holmes is apparently solving the case of the Red-Headed League. And the dialogue is performed by? Basil Rathbone.
Dumbo appears as a bubble-blowing toy in the toy shop that Basil and Dr. Dawson go to.
Clock Punk: The robot double of Queen Mousetoria that Flaversham makes for Ratigan.
Clock Tower: The Westminster Clock Tower, home of Big Ben, is the location for the final showdown.
Conspicuous CG: One of the first uses of CGI (after The Black Cauldron) in an animated feature, traced from wire-frame graphics onto animation cels and certainly a far more conspicuous user than its predecessor — where The Black Cauldron mostly limited CGI to special effects and stuff (i.e. glittering stars, chroma-keying smoke into the scenes, etc.), The Great Mouse Detective made extensive and notable use of it in the interior of Big Ben. Very impressive stuff for its time.
Conspicuously Light Patch: Averted in a scene where Basil and Dawson are crawling up some pipes to Ratigan's sewer hideout. One of them bangs their head against a pipe cap after taking a wrong turn, and despite being painted like the rest of the background, the cap rattles.
Cue Card: Mr. Flaversham is forced to read off cue cards while operating the Queen Mousetoria robot.
Curb-Stomp Battle: The final battle shows that Basil doesn't have any opportunity against Ratigan in a one-on-one wrestle. A good thing that this is a genius battle.
Dangerously Genre Savvy: When Ratigan learns that Basil is on the case, he realizes that it is inevitable that the great detective will track him to his lair, so he sets a trap for him there. He also sets up a decoy trap in the bar that fronts his hideout so that Basil will easily avoid it and be lulled into a false sense of security.
Distracted by the Sexy: It's quite easy to miss, but Fidget gets his leg stuck in the floor because he is watching the dancer's in the bar scene.
Does This Remind You of Anything??: One hundred percent unintentional, but there's a moment where Basil points to a small circular hole in a pane of glass and calls it "our friend's point of entrance" while sticking Dawson's finger into it. He then shows how if you pull, the glass operates like a door, but still.
Dawson: Dash it all, Basil! The queen's in danger, Olivia's counting on us, we're about to be horribly splattered, and all you can do is lie there feeling sorry for yourself!
Dueling Movies: With An American Tail, another animated movie about mice coming out the same year (though it didn't come out until a few months afterward). The Great Mouse Detective lost in the end, but that didn't mean it wasn't successful; in fact, it convinced Disney that its animation unit still had a future.
Establishing Character Moment: Basil doesn't appear at all for the first ten minutes or so of the film, letting the plot be established without him. When Dawson and Olivia are at Basil's flat waiting for his return, the movie (and the audience) is completely thrown off by the abrupt entrance of an insane-looking Chinese mouse, brandishing a revolver and screaming "I shall have him!". No sooner do we adjust to this intrusion that we learn that the Chinese mouse is Basil, wearing a realistic mask and a fat suit. But even after this, Basil utterly ignores the entreaties of the heroes to obsessively test his theory on matching bullets, howling in anguish when he is proved wrong and slipping into a spell of depression. It is only after all this that Olivia is able to get a word in and set the plot into motion, but by now we already know that Basil is not your typical Disney hero.
Besides his Villain Song, Ratigan's debut establishes his character quite easily; while threatening Flaversham, he speaks very tenderly about what could happen to his daughter, gently holds the toy he made for her....and squeezes it until it breaks.
Evil Is Hammy: Ratigan. Again, look at who voiced him. It's inevitable.
Executive Meddling: Largely averted, as Jeffery Katzenberg had little to no faith the movie would be any good. Most of his attention was spent on meddling with The Black Cauldron; we all know how that film turned out. Katzenberg did, however, demand the title of this film be changed from "Basil of Baker Street" to simply "The Great Mouse Detective". Two of the four directors have since said they've always hated this new title.
However those directors may have had some solace in the title being changed to "Basil the Great Mouse Detective" is some countries, including Australia
Face Palm: Basil during "Let Me Be Good To You" after Dawson starts making a fool of himself.
Faux Affably Evil: In song, Ratigan professes that he has planned his worst crime yet. His mooks gush "Even meaner? You mean it? Worse than the widows and orphans you drowned?" The thing is that his thugs asked the question with enthusiasm, and Ratigan was simply having fun the entire time.
Fluffy the Terrible: Felicia, a giant house cat that Ratigan flowers with affection and uses to dispatch his enemies as well as any incompetent lackey.
Furry Confusion: The movie has anthropomorphic mice, rats, bats, and lizards, but real cats, dogs, and horses, and humans as well.
Not so anthropomorphic, as Ratigan proves.
Genius Bruiser: Ratigan prefers to use his brain to solve his problems (if nothing else to preserve his image), but he is vastly stronger than anyone else in the movie.
Getting Crap Past the Radar: The stripping mouse named Miss Kitty, with this line "Hey fellas/I'll take off all my blues!" Everything from her bow to her shoes is blue.
"Hey fellas, / There's nothin' I won't do, just for you!"
"So dream on, and drink your beer. Get cozy, your baby's here!" Basil also points out his drinks are drugged. The fact that half the bar is smoking seems to be the least of the worries during this scene.
"Boys, what you're hoping for will come true / Let me be good to you!"
Russian version of the song is even more straightforward, with lyrics like: "I want to be tender to you, I want to love y'all!"
Although he does slowly grow out of it, as seen when he tries to comfort Dawson. Granted, he doesn't actually apologise for his rant, but it's clear from the look on his face that he regrets being so harsh.
Good Smoking, Evil Smoking: Basil smokes a pipe while Ratigan smokes a cigarette (complete with long stemmed filter a la Cruella de Vil). When Basil disguises himself as a thug, he swaps out his pipe for a cigarette.
GPS Evidence: The brandy, coal dust, and salt water on Fidget's list.
Heroic BSOD: Basil undergoes a near-fatal one after he falls for Ratigan's trap.
Homage: "Elementary, my dear Dawson". The entire movie, really, especially one scene where the silhouettes of two men who are obviously Holmes and Watson are visible discussing a case. The Holmes expy even shares a name with probably the most famous actor who played Sherlock. (So would that make this a Holmsage?)
In fact, in The Adventure of Black Peter, Holmes mentions he's known as (Captain) Basil around the Sumner area, though that much is coincidence. There's also Toby the dog, whom Holmes used in The Sign of the Four.
I Am Not Weasel: The professor HATES being called a rat; he's actually a "big mouse".
Idiot Ball: Basil and Dr. Dawson grab it pretty hard in the toy store by not realizing that someone had to have set off all the toys' mechanisms and that they didn't just wind themselves up. Basil should have instantly realized that whoever turned them on must have done so to provide cover for himself, and Dawson messed up by just letting Olivia wander off by herself.
Insistent Terminology: "I AM NOT A RAT!" note Ratigan has five fingers on his hands, er, forepaws, while all the other real mice have four fingers. Not to mention his yellow eyes, sharp teeth and thicker tail. This is different from the books, where he actually is a mouse.
Kidnapped Scientist: Flaversham is a toymaker, but he qualifies. His skill with making mechanical toys had applications for chicanery that Ratigan made forcible use of.
Knighting: It's certainly implied by the picture in the newspaper article at the end.
Large Ham: Ratigan is one of Disney's largest. In fact, Vincent Price said that this was his favorite role, and that he had a lot of fun with it. What's more, he also professed to being "flattered" that all of Ratigan's songs were written specially for him so that he could act as over-the-top as possible.
Mr. Price was prone to making wild, over-the-top hand gestures and such while acting, which the animators worked into Ratigan's character as well.
There's also the incredibly expressive, hyperactive unless quite visibly depressed, always overdramatic Basil himself.
Barrie Ingham (Basil's voice actor) was also given to dramatic gesturing. The animators had lots of raw material to work with!
Basil's most notably hammy moments are:
"Young lady, you are most definitely not accompanying us! AND THAT IS FINAL !!!
"Set it off now. Set it...off...now? . (laughs madly) Yes, yes, we'll set the trap off NOW!!".
Last Name Basis: (Padraic) Ratigan, (David) Dawson, and (Hiram) Flaversham.
Leitmotif: Basil, Dawson and Olivia all have their own respective themes that frequently show up in the film's score: Basil's theme is also used as the main title music, Olivia's is also the tune her father's dancing music box mouse plays, and Dawson's theme prominently features a bassoon.
Ratigan has two themes; a solo clarinet and a dark french horn/bass combination.
Lucky Translation: Basil is called "Basil Holmuis" (hole-mouse) in Dutch. Yes, that's a real word.
The Italian title for the film is Basil l'investigatopo, a play on words with "investigatore" ("detective") and "topo" ("mouse").
My Parents Are Dead: Basil, depressed about failing to catch Ratigan yet again and playing the violin to console himself, dismisses Olivia's request that he find her father with, "Surely your mother knows where he is." Olivia responds, "I don't have a mother," and Basil's playing comes to a screeching halt.
Red Eyes, Take Warning: If Ratigan's body starts to quiver uncontrollably and his eyes become blood-red, START RUNNING FOR YOUR LIFE!
This is used as a blink-and-you'll-miss-it bit of foreshadowing early on in the film. When we are first introduced to Ratigan, after he has just completed his Villain Song, he turns to Fidget, asking if everything is going as planned. When Fidget nervously reveals that he accidentally led Basil onto their master plan, Ratigan momentarily flies into a blinding rage, his eyes glowing a deep, bloody red, going so far as to sic Felicia on Fidget. But once Ratigan realizes that he can use Basil's involvement to his advantage, his rage subsides, but the redness in his eyes lingers until the next scene.
Maybe so he could mention Felicia? Not only was she his beloved gaurd cat, a ra-mouse taming a cat is a pretty remarkable feat, one which I would assume he'd want to mention somewhere.
Second Face Smoke: A woman at the tavern blows smoke at Dawson when he tries to apologize for bumping into her.
Shoo Out the Clowns: Ratigan dumps his lackey Fidget (who provided most of the film's comic relief) out of his dirigible, into the Thames River. That's a big clue that things are about to get dark and intense. Cue Villainous Breakdown and Nightmare Fuel inside Big Ben.
Sissy Villain: Ratigan (before he turns into a psychotic, feral rat at the end) is quite possibly one of Disney's most cold-blooded villains... and yet he surrounds himself with pink and purple fabric, fashionable capes, and sings songs while being carried around by his hired boys.
Staggered Zoom: Done on Ratigan right before he goes nuts during the Big Ben scene.
Steam Punk: The robotic Queen Victoria duplicate and its control system that Flaversham built out of cobbled-together gears and toy scraps commonly found in the 19th century, as well as Ratigan's favorite mode of aerial transportation, can only be explained as this.
Thunderous Confrontation: As the standard for Disney animated films, Basil and Ratigan's big fight on Big Ben near the end is accompanied by a strong thunderstorm, which clears up once Basil and Ratigan fall off the clock, and Basil survives thanks to using a piece of Ratigan's getaway blimp.
Title Drop: The last line of the film. The original title would have been "Basil of Baker Street", the name of the book series. Characters utter "Basil of Baker Street" quite frequently throughout the film, so if Executive Meddling hadn't changed the title, there would have been a lot of title drops!
Made even more frightening is the fact he was previously so Faux Affably Evil. It serves to show just how insane and feral his breakdown has left him, especially since he had, up to now, succeeded in keeping Olivia and her father separated from each other. When he sees Basil successfully reuniting them before his very eyes, this accelerates Ratigan's breakdown.
Worthy Opponent: Subverted. When Basil confronts Ratigan and starts speaking, it looks like he's invoking this trope, but it's actually the lead-in to an insult:
"Ratigan, no one can have a higher opinion of you than I have... and I think you're a slimy, contemptible sewer rat!"
Wretched Hive: The seedy pub. They DO serve Rodent's Delight, after all...
Writers Cannot Do Math: Really, Basil... just what is "the square root of an isosceles triangle", exactly? (Maybe Basil just is so much smarter than us that he knows how to calculate the square root of an isosceles triangle!)
You could also argue that he's thinking so fast that he's not verbalizing every step of his mental process, and there's some stuff in the middle that he just skips.