it really depends. If im on a long train ride, or some other situation where I'm sitting there I obviously would rather play engaging games on my nintendo ds, but if I just have a few minutes of free time on a short car ride or while waiting for someone i prefer games with small stages where i can beat a few quickly. When I'm at home I play on my console or pc, and my ds and itouch recieve no love except if I recently bought an exceptional ds game or really addicting itouch game.

I totally want a corny $1-2 app game like Angry Birds.

/sarcasm

I like them to be Pokemon.

...I'm leaning more towards the engaging side really. I use portables at home a lot and don't generally bother playing them unless I have at least a good half an hour of time.

My definition of engaging varies from most, though. I found Nintendogs to be quite engaging.

engaging!!! all i need is bejeweld for my quick game fix and PSP provides me with that too, 

I like engaging games.

But the metric of what is 'engaging' can be questionable. I found Game Dev Story to be engaging. I put ~60hrs into it.

I like more engaging games, especially this gen when the DS and to a lesser degree the PSP are 'the JRPG platforms'.  Though, due to the nature of handheld gaming, I like it when games have a 'save anywhere outside of battle' option, like in Pokemon, TWEWY and Golden Sun.

You asking this on a website that is going to vote overwhelming one way and thats to whatever fits Nintendo or Sony by your choices. I voted see results because I vote both, which the DS,gameboys and PSP offer. For example there was the Game and Watch series which was a lot of "15min fixes", but is on a Nintendo handheld. Theres a lot of examples of games that can be taken in both ways. It all leads back to why games cannot be considered hardcore or casual. They are just games made for us to enjoy ans as long as they do that they have done their job. I would also add I dont game on my phone or play games like the Facebook ones on my CPU. I use my CPU to visit forums and do my powerpoints/papers/ and other projects on. I use my phone to get in touch with someone. I use my DS and PSP to play games and nothing else.

I'd like the game to take advantage of the platform for instance.

The World Ends with You uses the touch and dual screens and makes a very unique game. That I wouldn't want to move from this platform.
As well as games that are easy to pick up and go play with wifi with friends in the same room (Playing in the same room beats over the internet anyday for me). (Mario's good everywhere).

However Games like
The 3rd Birthday
And MGS: Peacewalker(My 3rd favorite game this gen)
AKA Shooters 

I love both of those games. But they both would have easily benefitted from better controls on console. They both do a good job with what they have. Although, the 3rd birthday is the best SE shooter I've played (that's not saying a lot) despite being stuck to the psp controls. unless you count einhander... but they're different types..

Examples of in between
Kingdom Hearts Birth by sleep. Obviously my favorite battle system in kingdom hearts. probably made to lower dev-time and cost and works fine on psp. But would have rather had snappier load times, better graphics.
Crisis Core as well.

To me the god of war games are inferior to the home console versions so I'd nix against that.

Pokemon I think would work well either way... however, they're not going to do a full fledged home console game unfortunately.

To sum it up easily. I want the type of game to be on the platform that fits it the  best.

My favourite handheld games so far - VC, Jeanne d'arc, Tactics Ogre so i guess that means i'm not the guy looking for 5 minute long handheld games ;)

Win...if it weren't 'quick'

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:P

OT: I LOVE big engaging games! GOW, VC2, Dissidia etc. But there are simpler but yet good and engaging games too such as EXIT or Layton