I also like the Premier Ball for being literally a re-colored Pokeball. The general difficulty in acquiring it means you want to conserve it for whatever you deem to be a special occasion -- not unlike the Master Ball.
My favorite are the regular Pokéball, the Fast Ball and the Cherish Ball. I like how the Pokéball looks, it's simple. The Fast Ball looks cool, and I like the lightning design on it. I like the Cherish Ball because it is rare and when a Pokémon is sent out of it, the animation is colorful and pretty.
In terms of function, my favourites are the Quick and Timer Balls. Quick is helpful since it will literally never fail on a large number of Pokemon provided you use it turn 1, and Timer is great for catching things like Legendaries, since the catch rate goes higher than any other ball bar a Quick on turn 1, or a Beast Ball on a UB.
In terms of just general aesthetics, the Beast Ball's kinda cool, with its not-quite-circular design, and on a Pokemon like Porygon the effect becomes pretty cool. I think my favourite though is probably the Level Ball. Something about the red/black/orange design just clicks for me. Kinda sucks you can't get more than 1 in SM though.
If we're going least favourite too, I'd probably say the Fast Ball. The sprite looks kinda cool with the orange shade, but when the game renders it it just looks like a crappy painted Pokeball, and the red on yellow clashes too much for me.
Oh man sit down imma break down some stuff for y'all. For the most part I love balls solely on appearance. In no particual order:
Moon Ball: I LOVE CRESCENT MOONS, Y'ALL. Love them. One of my favorite generic shapes. I want them to make them purchasable in a new game so badly. Make them pricey if you must since they're 'custom order' stuff from Johto. I don't care. Lemme have my moon balls. I'm not using the one I have in Ultra Moon because I don't wanna lose it. The only way they could make me love this pokeball more is if they made the colors plum and black instead of the awesome color they already chose.
Love Ball: Super cute with the pink and the heart motif. If only it weren't only useful for pokemon of the SAME species of the opposite gender.. Like make it 4x effecting on non-same species pokemon of the opposite gender too.
Premiere Ball: I love the simple red on white. I always buy pokeballs in batches of 10 to get at least one or two of these every playthrough. I even managed to catch something with one in Ultra Moon: the Smoochum that became the Jynx in my champion team.
Dive Ball: I love the color blue and the blues and white in this ball just freaking works. I want to catch every single water-type in a Dive Ball.
Luxury Ball: I love the colors on this pokeball but I can never manage to catch anything I think would work well with the design. I wish we could buy pokeballs and then take them to the pokemon center and change out pokeballs on the pokemon we have (materball obviously wouldn't be an option but still). I would love to put my starters in luxury balls since they're always in my party and they're special to me.
Heal Ball: I love the colors on this pokeball. I also love that it heals your pokemon when you catch them. Thanks to the new mechanic in Sun and Moon it makes it actually worth using once you have a full team!
Dusk Ball: Its the only pokeball I know the release animation for and it's awesome. If only it were purple and black with a red line instead of green and black with a red line... I also play more often at night so that means I get a LOT of use out of these babies. (Curse you, Ultra/Moon for making my time inverted!!)
Dream Ball: It's so pretty you have no idea how badly I wish I could buy these in the games. I never got to have a pokemon in a dream ball. Just make them as effective as a regular pokeball and let us buy them. Or better yet, allow us to customize generic pokeballs at pokemon centers/special stores to allow us to take the generic white and red ball and select from a handful of colors/designs and have the Dream Ball and Park Ball among the options for recolors.
Shout out to the Rocket, Magma, and Aqua Balls from the Trading Card Game. They pretty.
Favorite pokeball is probably cherish ball with premier and master being second. I just like their designs and I’ll always associate a master ball with mewtwo whose my favorite Pokémon.
Function wise the timer ball or regular old pokeball. The timer has been way helpful in catching Pokémon I’m stuck in long battles with. Normally legendaries. And I swear the pokeball works better than the great or ultra most of the time.
Design wise, Timer Ball. First time I saw one I thought it was so cool, and I like the animation it uses when you use it. To catch a Pokemon I don't really have a preference actually, I just use what's best for whatever situation I'm in.