If you're looking for the episode where Luffy first uses Advanced Conqueror's Haki (coating his fists/attacks without touching the opponent), that is Episode 1028 (during his second round of fighting Kaido on the rooftop).
However, if you mean the first time Luffy uses basic Conqueror's Haki to knock out weak-willed people, that is Episode 413 (at the Sabaody Archipelago, after the Straw Hats are separated by Kuma).
But based on common fan discussions, most people asking about "Supreme Haki" mean the advanced, coating version. So the clearest answer is:
first uses Supreme King Haki (Conqueror's Haki) subconsciously in Episode 389, titled " Unveiling the Secret! The Weapon of the Sunny: Gaon Cannon
". In this episode, he unintentionally unleashes it to stop the bull, Motobaro, during the encounter with Duval. The Evolution of Luffy's Conqueror's Haki which episode luffy use supreme haki
While Episode 389 is the first instance, Luffy uses this rare power several more times before finally mastering it:
In the One Piece series, Luffy's use of Conqueror's Haki, also referred to as "Supreme Haki," is a significant plot point and showcases his growth as a character.
Luffy first demonstrates his Conqueror's Haki during the Marineford Arc (Episode 388 and 389, and more notably in Episode 391) when he is trying to protect his friends and particularly when he's fighting against Donquixote Doflamingo's subordinate, however the most pivotal and clear demonstration is during the Fishman Island Arc doesn't occur but rather during the Dressrosa Arc.
However a more clear showcase of Luffy's Conqueror's Haki is seen at: If you're looking for the episode where Luffy
During the Dressrosa Arc, Luffy uses his Conqueror's Haki to knock out the St. Charloss who was bullying the people of the towns, implying a much more controlled and deliberate usage.
The term "Supreme Haki" isn't commonly used; fans and the community sometimes refer to Conqueror's Haki by this term due to a mistranslation in early fan translations and English dubbed versions, which referred to it as "Supreme Haki" due to early inaccurate renderings.
The use of Conqueror's Haki is extremely rare, as it signifies a user's ability to intimidate and dominate the spiritual domain of others. Luffy's utilization of this powerful form of Haki plays a crucial role in demonstrating his strength, resilience, and capacity to protect his crewmates and others he cares about.
Luffy first utilizes Supreme King Haki in episode 389 against Duval's bull, with his first major display occurring in episode 413 to stop the Boa sisters on Amazon Lily. Later, he exhibits mastery post-timeskip, knocking out 50,000 enemies in episode 554, and eventually masters advanced infusion during the Wano arc. For a full, detailed breakdown of these episodes and Haki's, visit the One Piece Wiki. During the Dressrosa Arc, Luffy uses his Conqueror's
This is likely what you mean by “using Supreme Haki” as a fighting style.
If you want to see Luffy use the Advanced form of Supreme Haki (coating his fists without touching the enemy), the specific episode is:
If you want to see the first time Luffy uses Basic Supreme Haki to knock out fodders:
It is important to distinguish between "Basic Conqueror's Haki" and "Supreme/Advanced Conqueror's Haki," as Luffy has used the basic version much earlier in the series.