I used a Master Ball to catch Palkia, but that was before I knew Uxie, Azelf, and Mesprit were legendary.
It all depends on the types of the Pokemon or how long you expect the battle to last.
For Water-type or Bug-type legendaries, use Net Balls.
For long-lasting battles, use a Timer Ball.
For cave or night battles, use a Dusk Ball. (I recommend this for the Generation IV legendary trio, Uxie, Azelf, and Mesprit.)
For Water-types, use a Dive Ball. (I'm not sure where you get this.)
An overall ball to use would be the Ultra Ball.
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Any pokeball can do it but they each have a specific catch percentile or certain effect to boost their percentile but the masterball never fails at a capture.
You buy them in the Pokemon markets.
You can use any Pokeball to catch any Pokemon. Since Lickitung isn't a legendary or a 'running Pokemon' do not use your masterball. I recommend using Ultra Balls during the day, but if it is nighttime, use Dusk Balls.
You can buy them at the Pokemon mart at the Pokemon league. I buy my timer balls, heal balls, etc. at the Pokemon mart in canalave city
Cherish balls are the poke balls used to contain Pokemon obtained through Nintendo events, such as the Jirachi that was recently distributed at Gamestop.