Growing up, my first gaming system ever was a big gray brick GameBoy, bought in 1992. We got a second GameBoy the next year.

Got a GameBoy Color when it released in 1998. Sold at a yard sale when I got my GBA.

Bought a GameBoy Advance in 2001. Sold it some time later.

Got a free GameBoy Advance SP when some kid left it at my work in 2006.

In 2007, I got married and my wife had an original GameBoy. Then, my wife and I both got DS Lites.

In 2008, I got another DS Lite. Later sold/traded one to a friend for a DS Phat.

In 2009, all three DS systems were stolen. Same year, I bought a used GBA for Pokemon.

In 2010, I got a DSi XL, then later, a new DS Lite.

In 2011, I bought a 3DS.

In 2012, I bought a 3DS XL.

Altogether:
-GameBoy - 3
-GameBoy Color - 1
-GameBoy Advance - 2
-GameBoy Advance SP - 1
-DS - 1
-DS Lite - 4
-DSi XL - 1
-3DS - 1
-3DS XL - 1

Currently:
-GameBoy - 2
-GameBoy Advance - 1
-GameBoy Advance SP - 1
-DS Lite - 1
-DSi XL - 1
-3DS - 1
-3DS XL - 1

I plan to re-acquire the systems I no longer own (GBColor, DS Phat), and finish my collection with the ones I never did (GBPocket, GBLight).

I only have the GBA. Fun system but handheld gaming doesn't hold my attention very long.
I used it on the plane when I was still taking frequent trips. Once I got a laptop I used that instead.

One Gameboy Color
One DS Phat
One 3DS

They are all still alive and working properly, no damages whatsoever.

I grew up on a little village, without much contact with videogames, and I was so naive to the point to believe that the Gameboy was the pokemon hardware, I wasn't aware that it was a gaming platform with many other games to choose from. I ended up owning just pkm blue and silver. 

It was later, when I bought pkm silver that I understood the reality of a gaming platform lol. I was about 12 years old maybe. My previous consoles were those pirated ones that have 999999 games included on the main board, which consist of only 5 games and various (numerous) hacks of those games...

I've had every Nintendo handheld until the 3DS because it gave me a migraine within 5 min of playing it. My eyes can't handle the 3D aspect.

GBA and DS Lite

@pray: then turn it off.

Game Boy Original
Gameboy Pocket
Game boy Advance
Gameboy Advance SP
NDS
PSP
PSVITA

1990 - Gameboy
2008 - DS Lite

On 2009 bought a DSi for my parents as an anniversary gift: one loves Dr. Mario and the other Advance Wars.

More @ my gaming history post

Everything except the Virtual Boy (which wasn't really handheld, was it?).

I count my daughter's stuff as mine since.....well, I bought them. Currently:
-Two 3DS's (one hers, one mine)
-One DSi (hers)
-Two DS Lites (one hers, one mine)
-Three DS Phats (all mine)

-Had a GBA launch model w/ no backlight. No idea where it is.
-Have an clear Gameboy that I bought in 1995. Last seen somewhere at my mom's house a couple of years ago. Only game left is Tetris.
-Had an original Gameboy. Sold it to buy SNES games back in 1992. Was tired of it, at the time. I didn't even have a SNES!

Well, what do you know? I never had a Gameboy Color. It just always looked like another Gameboy to me. At the time, I had no idea it was considered a different console. I also never invested in a Gameboy Pocket or any of the other variations on the original Gameboy. I just cared about the games.

My insane game buying habits didn't manifest until 1998 and didn't become obsessive until this gen.

Proof that Sony copies Nintendo!

(because the OP asks which Nintendo handhelds you own and you listed the PSP and PS Vita.  It implies that Nintendo made the PSP and the Vita.  That means that Sony copied the PSP and Vita from Nintendo.  Yeah, I know it's a lame attempt at humor but I fell asleep on the floor looking at VGChartz and nobody bothered to wake me.  I woke up at 4am and I'm not really in my right mind.)

Whoops didn't read the full title!