The best Melee weapon in the game is the Chaotic weapons. The Rapier is best for PvM, Maul is best for PvP.
If you don't want to spend lots of time getting 80 dungeonerring then the Godswords are the next best option. Armadyl and Saradomin are joint best. The decider is that Saradomin Godsword heals you with its' special and Armadyl hits higher with its' special.
The best ranged weapon depends on how your going to use it. Dark bow is only good for PvP. Rune Crossbow is ok for PvP and good fo training. The chaotic crossbow is not reccomended for anything. Knifes and darts are good for training but chins are best for training (see guides on how to use them).
The best Magic weapons is Zuriels staff but it degrades. I would suggest Zuriel only for PvP and the Master Wand for PvP or PvM. Staff of light is good if your fighting a Melee person. The special stops 50% of melee damage agaist you.
Any of these weapons can be the best, it just depends on how you use them.
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In my opinion the best is a rune scimatar
vestas sword, but the best weapon is chaotic maul.
The overall best f2p Runescape weapon is actually a range weapon - Maple Longbow (sighted). It has a range bonus of over 100. The best f2p degradeable weapon is the Gravite 2h. Both are Dungeonneering weapons and need 45 Dungeonneering levels.
In my opinion it is a mithril scimitar.
An Abyssal Whip, this weapon requires 70 Attack to wield, but this being the best one-handed weapon in Runescape also comes at a moderately-high price inside the Grand Exchange, especially to a new member.