Beowulf

You need to kill at least one enemy with each available weapon. Weapons can be stolen from enemies or found in weapon racks throughout the game. To pick up a weapon from a rack, press . To steal a weapon from an enemy, first grapple him and then press when prompted.

Please note that the list above is where I first encountered said weapons. They also appear later on the game but getting them on their first occurrence is a safe way to get them all. I also advise getting weapons in the area first and then clear them of enemies.

1-handed Sword from Weapon Rack
Commonly found at the beginning of the game. This will be the first weapon you find.

Double Headed Axe, 1-handed from Weapon Rack
First found on The King's Road after you have fought human enemies for the second time and after Grendel has tossed a corpse at your party from off screen.

Curved, 1-handed Sword stolen from from Enemy
I first came across this in the battle with the ambush party right after one of your thanes destroys the road block on The King's Road right after coming across the camp with the Double Headed Axe.

1-handed Axe from Weapon Rack
First found during The King's Road when you get the objective to defeat the Barbarian Horde. Move into their camp. You will find this weapon on a weapon rack to your right.

1-handed Bone from Weapon Rack
First found on Grendel and the Golden Horn when you climb down the waterfall. Once you get out of the water there should be a weapon rack right in front of you!

War Spear chosen from Loadout Screen
You can choose this weapon at the start of Act 2.

Pitchfork from Weapon Rack
At the start of Innocence Sacrificed in the very first area there is a weapon rack on top of a small climbable area. Look for the blue shining marker.

Spine from Broken Pillar
During Beyond the Barrow you will have to hold off attackers while trying to close off doors. On the left hand side there is a pillar that can be destroyed in the middle of the way up to the left door. Destroy it to find this weapon.
Note The Spine is most likely a one-handed axe and, therefore, using it will count as using that weapon category. For more details, see below for what I suspect might be the way the game tracks weapon usage.

There might be more weapons but after using the spine to kill enemies I got the achievement. This leads me to the following observation. There are 1-handed slashing weapons (swords and axes) which share an animation. 2-handed weapons like spears and pitchforks also shared an animation and the spine I picked up last had an animation similar to a very heavy hammer one of your followers used at the beginning. I suppose those are the three weapon types you need to use.

Alternative (and by the comments, probably accurate)
If you check your Combo Book, accessible by pressing , you can see some categories. I suspect you need only kill an enemy with a weapon from each category for the achievement to unlock. I am however unable to verify this claim!

Update As by SiNNeR HeReTiC and his comment, Shield is a category in the Combo Book and as such you need to have scored a kill with a shield at least once.

Review 23 September 2014: Added button icon.
Review 6 April 2016: Missed a button.
Update 9 February 2022: Added information on shields

The Legend and Shadow weapons do not count towards this achievement. Simply go through the game and whenever you come upon a weapon you haven't used, grab it and kill an enemy with it. Also make sure and grab all the enemies weapon types and net a kill with them also.