The Stack Format
Apparently, "Stack" was already a format and has been discussed on Wizards' own website here and here. However, the decks I build are quite small by comparison, so I'll label mine as "Microstacks." The typical decks I build are 60-card with only two 100-card 4-player decks thus far.
Games can be played without having to bring multiple decks or having to count on another person to bring their own deck. This can enable pick-up games in numerous places where play is less likely to happen.
Originally this made it easy to play Magic between at school/between classes. I've also been able to get in games before FNM drafts when people are reluctant to break out large EDH decks with all the related paraphernalia when those games typically go on for much longer.
Format Quirks
Putting cards back into the library can change who draws them. Metamorphose is my favorite example. For and a draw step, you get to steal any permanent.
Manipulating the library with cards like Halimar Depths and Mul Daya Channelers can be used to steal better cards from your opponents, but they can fight back with cycling, dredge, or mill.
Cards like Unsummon behave as you'd expect them to, though I've yet to introduce Mind Control or Switcheroo, so I'm not 100% sure what my opponents would expect bounce to do in that situation.
Disentomb effects become deeper when you can target a creature card that you just killed. Evolution Charm is always a cool card.
If the Stack is mill-themed, it's a battle of Mill-To-The-Death!
Threshold will activate very quickly. I avoid instants with flashback. Dredge cards can be fought over. The graveyard is as relevant as you choose to make it.
I find that designing a Mircostack is essentially like designing a limited environment, though I am admittedly drawn to limited-level cards out of preference. Regardless, I notice dynamics and trends in my decks that I can actively fix. With the red deck, instant-speed removal turned out to make combat tricks terrible draws and I was able to correct that by taking them out. It was a very liberating experience removing Burst Lightning and Brimstone Volley for cards that would make the gameplay better.
I have noticed that I like to make the power level of cards relatively flat so that drawing a land is closer in power level to drawing a spell than it would be otherwise.
Normal games of Magic have two decks with their own goals. Some come out blisteringly fast and spend extra cards to do so. This match up typically has one deck that has better top decks with another deck that will probably lose if it doesn't draw the right hand.
My Microstacks don't have this dynamic at all. Each player has the same top deck quality, so a flatter power spectrum is needed to lessen the effect of draw variance in these games. It is also why I shy away from cards like Divination.
That's all I can think of for now, but it should give good insight into what I'm doing.
I also like it when more of the lands players draw are relevant. Either there are things for players to do with more mana. Or the lands turn into spells. Or the lands provide a rare splash color.
I prefer splashes to give bonus options on cards that have already been played, affecting as many cards as possible while still being reasonable cards without the added effect. I will allow few cards that outright require the color and such cards must provide extreme utility in the context of the deck.