Battle Moveset

A Good Moveset for Linoone would have to be this:

Attacks:

Flail
Substitute
Toxic
Pursuit

Items Attached:

Focus Band

Preferred Nature:

Impish

Strategy Using Linoone

Linoone has good speed, but its other stats are somewhat bad, but that doesn't mean it can't use that wide array of moves it can learn.
This moveset is supposed to annoy your opponent, Toxic your opponent's pokemon, hopefully this will trigger a switch, and if he does switch, use Pursuit.
Granted, Pursuit isn't such a powerful move, but it will deal that extra bit of damage when your opponent tries to switch, then you can toxic again.
With Flail, if your HP is reduced to a certain amount, it will become really powerful, with the boost from STAB it will more than make up for its low attack strength.
Substitute will force the battle to go on longer, allowing toxic to take effect and reducing Linoone's HP till it can use Flail. In case you don't want Substitute, replace it with odor sleuth, just to make ghost types vunerable to Flail.
The hold item, Focus band, even though it has a low chance of working, may be something that you'll end up having to rely on. Either that, or replace it with Leftovers.
Linoone is weak to fighting types, so with this moveset, it is still fairly weak to moves like that.

EV Corner:

Max out the EVs in Defense for Linoone with any remaining in Special Defense & HP

Strategy Against Linoone

Linoone' Defenses are equal and Just Below Average so any volley should get it in a turn or two. Especially Fighting Type.

Contest Moveset

A good Contest Moveset for Linoone would have to be this for the Cute Contest best with Timid, Hasty, Jolly or Naive Nature:

Attacks:

Belly Drum
Rock Smash
Flail
Covet

Items Attached:

Pink Scarf

Strategy Using Linoone

If you do the attacks in this following order you should have very few problems; 1st - Belly drum, 2nd - Rock smash, 3rd - Belly Drum, 4th - Rock smash, 5th - Flail/Covet

Locations in Games

Ruby/Sapphire/Emerald Routes 118, 119 (R/S/E), 120, 121 & 123 (R/S) Colosseum Trade from R/S Fire Red/Leaf Green Trade from R/S

Animé Appearences

Linoone has had a few Animé Appearences. In it, it was fleeing from the Fake Groudon that was ensnaring all Pokémon. It later appeared and stole some of May's Pokéballs Movie 6: Jirachi - Wish Maker