I've been doing research to find a crypto-currency that is "fairly" distributed among the population. Being "fair" is quite subjective. We can all agree that 10 people that control a coin is not fair, but what about 100? or 1000? or 10,000?
So the question is, which crypo currency or currencies have the most distribution? And the other question is distribution the same as adoption?
"Widely" distributed and "fairly" distributed are two completely different things.
"Fair" means that the mechanism by which the initial distribution was determined is fair, regardless of who ends up with more coins. You could have a fair coin which ends up wholly in the hands of one person, if he earned those coins legitimately and fairly. And you could have an unfair coin which ended up widely distributed. Eventual ownership is also distinct from initial issuing.
Narrow distribution could be evidence suggesting that the mechanism was unfair, of course. But the notion that narrow distribution on its own is unfair, contrasts the ethical position that people should be responsible for the consequences of their actions. If someone works hard and makes smart decisions to create and obtain value, there is nothing unfair about it.
Bitcoin is clearly among the fairest, without a privileged party and with continuous issuing over many years. I also believe it is the most widely distributed, measured by number of people who consciously own bitcoins. It may not be the most equally distributed, though.
Distribution is not the same as adoption. If you manage to issue a coin equally to every person on earth, then the distribution will be wide, equal, and arguably fair - but if nobody actually uses it or cares about the coin he was issued, you can't claim that it's widely adopted.
As you noted, a lot of this depends on how you define your terms. In an Objectivist sense, all currencies are equally fair. :)
If we look at fairness in terms of information disparity, we can see some clear winners and losers. Information disparity is important because if insiders know that their coin is technically sound before the general public does, they can snap up coins while they're cheap.
If we define fairness in terms of "one person, one coin:"
You also ask about which coin has the most distribution, which is a different question from which one is distributed the most fairly. In terms of active users, Bitcoin probably still has the most users. (Dogecoin is catching up, though.)
In terms of inactive users, AuroraCoin theoretically has 300,000 owners. (Of course, by that measure, you could increase adoption of your *coin of choice by putting money into private keys and emailing them to random people on the Internet.)
Clearly Bitcoin. Because it uses the only decentralized method of distribution known to man. Plus it has had the longest time to run and even under the radar for quite a few years (anything started now won't have that chance anymore).
According to this study of alt-coins done at a Princeton university seminar, the greater the market cap, the greater the distribution (and they do account for multiple addresses on one hand, through transaction graphs). As expected by the enhanced utility stemming from network effects, the larger the market cap, the more wealth is distributed, hence it seems Bitcoin overtakes Litecoin, which overtakes Dogecoin, etc.
Their source code is available in github.
As for the second part of your question, is distribution the same as adoption? That may not be the case; for many pre-mined coins (or insta-mined coins) have been widely distributed but had low traction. People just ignore them after a while (stellar, nxt, auroracoin, etc.)
