pac-man is an old standby, as is galaga, both verticle games.
I'm not saying Galaga is a bad game, but I've never understood the widespread fascination with it. There were dozens of vertical shooters in the same genre. Many of them were as good as Galaga, some were probably better, some were definitely more influential. I mean. . . Space Invaders, Astro Invader, Carnival, Galaxians, Phoenix, Moon Cresta, Radarscope, Pleiads, Satan's Hollow, Gorf, Astro Blaster. . . The list goes on. Back in the day, Galaga was merely one of the many, many such games on the scene. Somehow in retrospect Galaga seems to have been elevated above all the others, but I can't say why.
My personal favorites from that genre were Moon Cresta, Phoenix and Astro Blaster.
Some interesting variants on the whole vertical shooter idea included Centipede & Millipede, which used a trackball. Tac-Scan was a vector game that let you control multiple ships using a spinner. Quasar combined vertical shooter stages with Asteroids-like stages. And one of my all-time favorites is Gyruss, which is like Galaga wrapped around a Tempest playfield. All of these were vertical-monitor games, I'm pretty sure.
Vertical monitor orientation wasn't uncommon at all. In fact, from my list of old favorites I think just about half are horizontal, half are vertical.
Some other vertical-monitor classics were: All the Pacman games. All the Donkey Kong games. Ladybug. Scramble and Super Cobra. Tempest. Arabian. Arkanoid. Ikari Warriors. Moon Patrol. Rally-X. Zaxxon and Super Zaxxon. Vanguard. Xevious. Sinistar. Quix. Dig Dug. Frogger. Q*Bert. Space Dungeon. Time Pilot. Tron. Truxton. And I'm sure that's barely scratching the surface.
I've sometimes thought a vertical-monitor cabinet would be cool. However. . . If you play a vertical game on a horizontal monitor, it's 3/4 scale. To me that's acceptable, for something that's at arm's length. It's not like you're looking across a room at it. Say you have a 20" monitor. . . Your vertical games will be sized as if they were on a 15" monitor. I can remember when I used a 13" monitor with my computer for several years, and 15" monitors were high-end gear.
If you have a 25-inch monitor in your cabinet, which isn't really excessive by today's standards, then you can play vertical games sized as they would have been on a 19" arcade monitor -- which is authentic for most of those older classic games. To be really authentic you'd have to program it to scale down the horizontal games to match! But I can't envision most people wanting to do that.
Maybe somebody will make a square monitor someday, so it'll be equally inefficient at showing both types of games.
BTW. . . I once saw and played an original Quantum machine in its natural habitat. I've got to say, it wasn't that fun. Neat concept, just not fun to play. I wouldn't describe it as Tempest with a trackball, I'd say more like Qix with a trackball. . . but that's not completely accurate either.