Per wiki policy, Spoilers Off applies here and all spoilers are unmarked. You Have Been Warned.

Mission 1: Devil May Cry

The very first action scene in the Bad Guy Bar (and to an extent, this entire anime) has Dante shooting his Ebony handgun against a suspicious man who attacks him from behind. Dante doesn't look back while doing it, and the man's corpse flies backward in slow-motion. But as that man's body is still airborne and didn't fall to the floor yet, it immediately transformed into its real demonic appearance, and lunges back at Dante. Naturally, Dante quickly dodges. For an anime of its time, this is already an impressive fight choreography.
In the briefing scene, Dante makes the act of ending a telephone call look cool by tossing over the phone and making it land perfectly on its stand.

Dante couldn't catch up with the biker gang leader Vincent and stop him from chasing the motorcycle-riding demon "Red Eye" ahead of them. A direct shot didn't do much against the demon, and when Dante correctly guessed that the demon uses sound to make its victim hallucinate, Dante decided to destroy the bridge they're on by rapidly firing his guns at the bridge's railings, so much so that the bullets also ricochet all over. To top it off, the anime's main Instrumental Theme Tune, "d.m.c.", starts playing as the bridge is falling down.
During the motorcycle chase, Lady does a quick sideways-pointed Guns Akimbo after letting both her hands off the handles. Despite this, she still surprisingly maintains her balance.

While it's regularly established that many demons in this anime are disguising themselves as humans to blend within society, they still mostly attack by lunging, biting, or clawing their victims at close range. The Mafia demons in this episode are different, as they prefer to use guns instead, resulting to them fighting Dante in a shoot-out.

One that doubles as both hilarious and awesome is Trish juggling the skinny demon in mid-air using her handguns, just like what players can do in the Devil May Cry games.

The episode's title appears only when Dante smiles at Patty after having made it out of the underworld to rescue her from demons in the hotel. Then a guitar plays in the background, as the hero is not yet dead. "Stylish" indeed.