Does the choice reward or punish the player? How severely and how does it intersect with the moral context of the choice?
A moral choice where being good gets you x reward, being bad gives you x + a bit more reward.
Missed skin moral fork
A moral choice where being good means you miss a chance at something and it feels like it stings.
Lost skin moral fork
A moral choice where being good loses you something that will actually sting. The distinction is important: there's a psychological quirk where people feel the loss of something they possess much more than the potential loss of something they could possess. Even when the net cost is the same in the larger context. Really upsets economists. :)
Missed blood / lost blood moral fork
As above, but it's something really rare and important.
Flagged moral choice fork
As any of the above, but it flags you with personality qualities.
Implicit moral choice fork
As any of the above, but it makes you think about what the best thing to do was. But all this is purely 'a game in the Pope's head' - it doesn't have any different effects. (A 'pure reflective choice', aka a 'reflective choice with a twist of cheat.')
How is the morality of the choice portrayed? How clear, grey or concealed is it?
Kittens moral fork
A clear moral choice, as in, do you eat or free the kittens?
Kittens triple fork
A clear moral choice with a safe neutral way, as in, do you eat, sell or free the kittens?
A Moral Maze choice. Is censorship always bad? Should the murder victim get to decide their killer's punishment? &c.
A showily painful moral choice. Eat the baby and save the mother? Or save the baby and eat the mother?
You're not given enough background information to be really sure what's going on. Eg, the investigation choice in Fallen London where you can leave the devil with his lover or not, but you don't know whether he's truly reformed. A bit of a cheat, or something troubling and interesting? Could be either. cf personal, covered fork.
It's really down to how you feel about a situation. Help the Tomb-Colonists or the Constables? Who do you prefer? Let the reformed criminal go?
A personal fork where the player has a reasonable basis or cover for claiming that their personal bias is supported by the facts of the fork. "It's obvious she's up to something, so I shot her," when in fact it isn't particularly obvious or even necessarily implied. Strong overlap with murky fork. The infamous Comtessa story in Fallen London probably qualifies as Covered.
It's really possible to construct a strong argument for either case.
Fork 1: will you spray graffiti for the cause? Yes? On to fork 2! Steal money for the cause? On to fork 3! Set fires for the cause? On to fork 4! And so forth. This kind of stakes-raising is a well-established dramatic tool and a rather nasty RL way of recruiting people into atrocities: if you say no, you may implicitly acknowledge that what you've already done was wrong. It's particularly compelling (or cheap) when there's a narrative someone's keen to see the end of.