Main observation: what to do to a recorded vocal to make it sound better depends on (at least) three things: (a) the recording you're starting with, (b) what other sounds are going on around it (presumably some form of musical accompaniment, but you never know) and (c) what "better" means to you. Which suggests that trying to answer the question is a bit of a fool's errand.
Another observation: if you've ever sat in a room and listened to a really good singer sing - just sing, without any mic or amplifier or anything - there's a good chance you thought it sounded better than any variation you could produce by fooling around with it. True, there is some "effect" being applied, in the natural reverberation of the room. In the "real" un-recorded world, some room reverb generally improves the sound. If you somehow convince the singer to go outside and sing in a field while you stand nearby and listen, it won't sound as good. Going further in the other direction, those of use blessed with voices that might charitably be described as "so so" usually sound a lot better (to ourselves, anyway) when we sign in the shower. I think that's why people so commonly do sing in the shower (along with boredom and a desire to distract themselves, I suppose).
In the same vein: karaoke bars typically have their equipment set up to drench the vocal in reverb. They also assume the minimum level of signing ability will be on the low side.
Going in a less abstract direction: in the real world, the effects that people typically put on recorded vocals are (in approximate order of frequency):
- reverb
- compression
- EQ
- double-tracking (which isn't so much an effect, as a recording method ... I'm talking about actual double-tracking here, as in singing the same thing twice)
- other delay effects, like slap-back or something more esoteric
- de-essing.
And, of course, another one: pitch correction. Let's put that one in a different category.
Typically, if sources are recorded dry / close-miked, there's some reverb on the whole mix, just so it sounds like it's something going on all together in some recognizable place. Whether you put extra reverb on the voice depends on a bunch of things. Generally - as in the bathroom and the karaoke bar - the worse the singer, the more reverb will improve things. It also depends on the style of music and the "sound" you're going after. Extreme examples: a rough-voiced, pitchy nasal folk singer with just a guitar might sound best very dry and "authentic," while a capable, smooth performance a la (say) The Supremes might sound glossy and full with a lush reverb (okay, perhaps not a perfect counter-example, as Diana Ross did have a pretty wispy and not-that-expressive voice).
Compression isn't so much about the vocal as the mix: the purpose is just making the vocal mesh. For a big rock sound, there's probably a lot of compression on the vocal to keep it in front while having the band sound loud. Of course, there's a lot of compression on everything, to make everything sound really loud. For a voice all by itself a cappella, you probably wouldn't use compression at all, unless the singer's volume is unintentionally wandering all over the place. "Unintentionally" is key word there: a vocalist's volume might intentionally be all over the place, which might be a good thing, depending on the type of music.
EQ is pretty much the same. It's used either to make the vocal fit into the mix, or to correct something that would otherwise sound wrong. The latter shouldn't really be necessary unless there's something funny about the voice or the equipment.
Double-tracking is highly useful on a not-so-great singer. It tends to even out the flubs and make pitch problems less evident. On a good signer - particularly a highly expressive one - it's less likely to help, and considerably more likely to hurt. You don't double-track Aretha Franklin or Otis Redding.
