Finn has used many swords. Each unique in origin and power. Some are epic, some are bland.
Regal Sword: Just a regular sword Finn used after buying it before sneaking off into a dungeon. The gray bulb was originally red until his mom ate the color.
Ice Sword: A Christmas present from Grandpa Simon. Ice King made it, adding a defending spell from the crown. This sword was unfortunately destroyed by Scorcher after accidently hiring him and the spell flew back into the crown. So, Finn went back to using the Regal Sword before getting the next one.
Abadeer Demon Sword: Finn gets it during the second time meeting Grandfather Hunson. (The first time, Hunson tried to bond with him by trying to suck all the souls in Ooo) Hunson gives it to the little "prince" before making the offer to Marcy in taking over the Nightosphere. Events go down normally, even Finn wearing the amulet. So, Finn gets a new sword and trauma that day. This dark blade is infused with Hunson's magic, giving it power to suck up souls, chop ghosts and overpower even the strongest of foes. Marceline, however, destroys this one after finding out that the Demon Sword was corrupting her son to do evil, finding out Finn's been sneaking out dungeon crawling even before he got the sword. Though the effect was taking way slower than Hunson expected for some reason. (*cough* Comet *cough*)
Candy Cane Sword: With the Evil Sword Curse still lingering on Finn, Marceline banned her son from going into dungeons. He went to the Candy Kingdom to look for any thrill, but not too extreme as he's still a good-hearted hero. He went to Princess Bubblegum to see if there's anything he could do. As Hero of Ooo, she saw an opportunity and gave him a job as a knight, giving him a new sword. Though he had a breakdown and quits after enduring trauma from the second zombie incident, but he still kept the sword.
Celestial Light: Seeing the curse becoming worse for Finn, Marcy feared she was going to lose her son like she lost her father-figure. She looked through the Enchiridion and the library to find at least something to help him, even a little. (Now, I didn't think of a full story here, but they end up in another dimension, earns the orb and it lifts his curse) Made of purifying light, Finn could shape it into any tool he willed it to with a proud shout of its name. Celestial Light was destroyed when it stabbed into the Lich's heart, erasing him and the blade from existence.
Biotech Sword: Finn gets this sword from when he, his mom and BMO somehow ended up in space... on the Drift. At first look, it's a cool looking sword, but it can adjust its effectiveness from able to block attacks to slicing anything like hot butter. It can also create arcs of energy with every slash Finn makes in close combat or ranged and release shockwaves with every hit if he wants to. If the sword has been knocked away or just out of reach in general, it will fly back to him with just a thought.
Soul Sword: Created by Peppermint Bulter before the Lich War, a soul trapper. Finn used it on the PepperLich to end to war and dumped it in the same volcano where Magic Man threw Patience St Pim after she used him as an elemental spell battery, but not before turning her into an avocado.
Biotech Arm Sword: Shortly after losing his arm, the Biotech Sword merged with the severed appendage and immediately reattached itself onto him, giving him a hard light blade with the same features as before. At one point after, he discovered it could customize its appearance, giving himself claws.
There was that one sword he made with the Sister of Flame (Flame Princess), but he never used it and gave it to her after she overthrew her father.
And that's all the sword MMAU Finn has used.
