Justin asked this very question not long back, this is the list I gave him then...
Bomberman 94, A great representation of a classic game. A 4 player adapter needed for multiplayer though.
Outrun, A simply ace and faithful reproduction.
Bonk's Revenge, More than just a Mario-a-like, it has some great innovations and puzzles to keep you interested. Infact the series is great fun.
SideArms: Hyperdyne, Another arcade conversion. Side scrolling shooting mayhem. This is double, double hard.
Tricky Kick, another innovative puzzler. Probably not so great if you have an objection to kicking cute animals about a screen though.
Circus Lido, A frankly bizarre and tricky puzzler where you play a Gecko chomping your way through a variety of bugs that imbue with abilities to solve each screen
New Zealand Story, Another great arcade conversion
Don Doku Don, A Bubble Bobble/ Snow Bros alike affair. Great fun and graphics.
Gunhead/ Blazing Lazers. Frankly, in the huge amount of competition for top down scrolling shooters on this system, this is the daddy. Bright, well drawn sprites, awesome power-ups and great fun.
Soldier Blade What I said above, sorry, total horlicks. I made a mistake, this is the daddy of them all. Bonkers gameplay, awesome powerups and great music to blast along too.
Galaga 90, Galaga, but with nicer sprites. I like my classic arcade shooters.
Chew Man Fu Another puzzler, a maze game where you push around coloured balls. Sounds dull, but it's not, it's great fun.
Coryoon Side scrolling shooter using a baby dragon as the main protagonist. Looks cutesie, but is also a satisfying blast-em up.
Hana Taka Daka Pretty much as above, but with a Goblin and ace power ups.
1943 Another Arcade conversion, top scrolling shooter faithful to the Arcade .
Photo Boy A system exclusive (I think) where you play a "paparazzi" type photog snapping pics of interest to your editor as the world scrolls in front of your eyes. Watch out for UFOs and starletts exposing themselves. A sometimes funny and very entertaining break from all the blast em ups.
Air Zonk. My absolute favourite side scrolling shooter, based on future Bonk. Total nutty shoot em up carnage with weird and amusing power ups and fantastic boss fights. Get this game.
Ninja Spirit Shuriken/ sword run and slash em up scrolling platformer with tough bosses and great game mechanics. I love this game.
R-Type Keith's favourite, and an almost perfect conversion of the original arcade game.