Within the opera scene itself, they trick the orchestra into playing "Take Me Out To The Ballgame"... and then start tossing a ball around.
Including using a violin as a bat.
On the steamship, Groucho is already dismayed to discover he's been given a broom closet sized room. He opens his trunk to discover Chico, Harpo, and his client Ricardo have stowed away. On top of it all, apparently every maid, electrician, carpenter, plumber, etc. is assigned to service their room at the same time.
Groucho's deadpan reaction to a very large engineer's assistant showing up to clean the place after he'd already had a number of other people cram themselves in. "Ah yes, I had a premonition you'd be coming."
The Contract scene
Groucho: It says the, uh..."The party of the first part shall be known in this contract as the party of the first part."
After shredding nearly every part ("The party of this part shall be known...") of the contract out:
Chico: Hey, what's this part?
Groucho: It's all right, that's in every contract. That's what they call a sanity clause.
Chico: Ahaha, you can't fool me! There ain't no Sanity Claus!
And the reprise, in which the contract-ripping is one-upped by Harpo ripping the back of Gottleib's tuxedo coat.
...and two hard-boiled eggs. *HONK!* Make that three hard-boiled eggs.
Chico: The first time, we get halfway across when we run out of gasoline. We got to go back. Then I take twice as much gasoline. This time, we were just about to land, maybe feet...when, what do you think, we run out of gasoline again. Back we go and get more gas. This time, I take plenty gas. We get halfway over, when what do you think happened? We forgot the airplane. So we sit down and talk it over. Then I get the great idea...we no take gasoline. We no take the airplane. We take steamship. And that, friends, is how we fly across the ocean.
"Was that three guys with beards, or one guy with three beards?"
The scene where the chief of police tries to prove that the other three are hiding out with Groucho, with them progressively hiding the evidence (four beds) when he's not looking, which only makes him think he's going mad.
Henderson: Say! What's that bed doing here?
Groucho: I don't see it doing anything.
Harpo's hijinks while the tenor sings "Mal reggendo" and "Di quella pira." They put a battleship in Il Trovatore!
When Harpo, Chico, and the opera's orchestra conductor get into a baton-tapping contest with each other.
Groucho: Not that it's any of my business, but I think there's a brace of woodpeckers in here.
A performer uses a whip to tear off the skirt off a female performer, leaving a smaller skirt. Harpo uses another whip to tear the smaller skirt off of the performer, leaving her in tights and underwear. Groucho notes to someone in the audience, "Now we're getting somewheres!"
Especially because Harpo then goes on a spree and just randomly rips the skirts off of the rest of the actresses.
Groucho's take on a gypsy woman in the opera: "Boogey boogey boogey!"
Groucho: They're the greatest aviators, but you'll notice they're traveling by boat.