This list is subjective, bossily opinionated, non-exhaustive, and covers only 3D MMOs. It focuses on current and playable titles.
Read my other blog posts.
Join a guild; almost all of them can be described as "mature and friendly."
Unfortunately there are no RP guilds anymore. They died off due to the failures of the game's former owner Gameforge, the release of Vanguard, and a virally marketed heavy-RP MUD. Connect with individual roleplayers: Marelli, Lomilmalia, Suibom, maybe Zyratuan; Pero and Riveit are good for political conversation; Drakfot is a famous storyteller; Sherkalyn, Rikutatis (who runs a nice, active Atysian newspaper) and a few others are part-time RPers; there are others I haven't met or who have returned without my knowledge.
(Updated on May 8, 2010.)
2. EVE Online
The premier sandbox. If you're one of those people who are put off by the lack of a human avatar, play anyway. Incarna a.k.a. Ambulation a.k.a. Walking In Stations is coming within a year or two and that gives you plenty of time to integrate into the community, develop your skills, and amass some wealth. If you're a roleplayer, look on the official forum for RP chat channels and join them.
3. Myst Online
The most story-driven and social-centric MMO in history, I kid you not. It's not actually playable right now, because it sort of died, but t's going to return eventually as an open-source project.
Update: Now it is playable. There is no fee to do so, but if you appreciate the game, please put your money where your mouth is by donating.
4. Wurm Online
I have little praise for this one. There's no RP, unless that has changed since last I played. Actions like mining take about 45 seconds, which is frustratingly just long enough to make you realise you're wasting moments of your life but just short enough that you can't web-browse or read a book meanwhile - kind of like Istaria's harvesting. Some of the textures are atrociously unprofessional beyond all belief and there's no third person camera, but...somehow, the environmental atmosphere is realistic and relaxing. You can make your mark on the world uniquely with terraforming, semi-freeform construction, farming, arborism and woodcutting, et cetera.
5. Fallen Earth
A post-apocalyptic sandbox that just released. I haven't been following it closely and I have only played its Halloween trial, so I can offer little comment. Heard that it pulled a somewhat Vanguard-style launch and my experience (plus a glance at the game's forum) corroborates that. Don't let that daunt you, however, because between the rough edges is a proper skill-based craft-centric sandbox, and it continues to be polished.
Indie, open-source, RP encouraged. Definitely not your average bear.
7. Istaria: Chronicles of the Gifted (formerly "Horizons: Empire of Istaria")
Don't let the fanboys fool you, the crafting system sucks compared to Ryzom's. It may be broader but it has the depth of a pancake and the the grind is mind-numbingly horrific.
But the game is a sandbox, more or less. There is even non-instanced construction.
AND YOU CAN PLAY A GIANT FLYING FIRE-BREATHING LAIR-DIGGING DRAGON. Plus ten biped races.
Ranked last because of strange management. Virtrium isn't at all fraudulent, just arrogant and annoying. Your mileage may vary.