Which of Microsoft's three home consoles do you think is the best, and why?

Xbox 360. Largest library of games, first console to market, great 3rd and 1st party titles, stable XBL, lots of new features, and much more.

360, multi's were superior.

Xbox 360. It has some memorable exclusives plus a huge amount of games, the first iteration of achievements (also the best), an awesome interface (until MS ruined it at the very end, but at least vanilla and v.2 were damn great) and brought a comfortable controller (shame about using batteries, but you could always buy one of those wired ones). It also had some smooth online interactions and allowed for a good experience overall.

It had its flaws, RROD and certain other issues, but I personally love it. It was my favourite seventh gen console and one of the best consolles overall imo. It also had some unique charm, can't really describe it. Something the Xbox One is lacking, and I'm not sure MS would be able to recapture it ever again. It was also a different era back then, but still...

easily the 360

Xbox360, only xbox I have.

That's tough. The First Xbox introduced online play which was unheard of for the tv. Ridiculously fond memories with friends for Halo 2 so that alone carries a lot of weight.
The 360 introduced one of my favorite franchises, gears of war which also stacks heavily in my mind.

I'd have to edge it to the 360 even though there were more games that I felt I enjoyed with the first Xbox (psychonauts, republic commando, battlefront 2, halo 2, the first Fable)

While I do think the original Xbox is underrated I'll have to give it to 360 for the huge library of games and superior controller.

I'd rank them:

1) X360 - a lot of great exclusive content, superior third party games, great controller
2) Xbox - clumsy controller aside, Xbox supported lots of great PC ports and two great Halo games
3) Xbox One - third party games are now largely inferior and Microsoft studios are closed or underperforming

Dude, you're reading my mind! I couldn't agree more, and I absolutely love Psychonauts and Republic Commando.

OG Xbox has a shaky reputation but I love it :) 

Xbox 360 was a great console.