Your first two Prophets should be used to make a religion & then enhance it. Any after that get one of two uses: convert cities that are putting out pressure against your cities, or drop them as Holy Sites for added

+ later

/tourism.
Scientists generally get turned into Academies early on. The +

per turn is often more valuable than popping them to speed up a tech. You can also drop them on strategic (
not luxury) tiles to work them, and still get the +

boost. Later on, you have a choice: if you're behind on

/turn, keep making Academies. If you're producing plenty of

/turn, just pop them when you have a tech with a long research time.
Generals have a minor buff to your troops if you keep them close, but that really hinders your tactics. You're better off using them to make Citadels. Preferably put them at a corner of your borders so that they gain you vital resources and/or the most tiles. This
will upset your neighbor if you claim some of their territory, but sometimes that's the fun part.
Admirals really only have one purpose: keep them close to your navy and pop them when you have multiple ships with damage. The insta-heal is their only practical use.
Engineers are worth holding onto if you expect to have a new strategic resource soon, either by expansion or by conquest. Also, if you expect a new one through research (ie. coal, aluminum), it can be worth it to keep an engineer or two in reserve. Otherwise, drop them on any tile for +

. If you're tight on

, drop them onto a hill, even if it's already got a mine built. That way, you can at least get the bonus

.
Merchants should be sent to a city-state you need to improve relations with, and popped in that CS's territory. If you're tight on gold, or just don't care about CSs, drop a customs house. Again, make sure it's on a
strategic resource, or on a blank tile.
Artists should basically always be turned into Great Works. The only exception is if you lack enough buildings to house the art; then it might be worth it to just pop a Golden Age. If you're already in a Golden Age, or working on a Culture victory, hold the Artist until you
do have room for the GWoA.
Similar with Musicians: turn them into Great Works unless you lack the room, or need to culture-bomb another civ. Note that you can only culture-bomb if you either have an open-borders agreement with the civ, or are at war with them (which means your musicians need a military escort

). So, most of the time, you're better off parking them until you get room.
Writers should always be held until you have room for their Great Works. The Political Treatise boost to

isn't really worth it unless it'll push you into your next policy
and you really, really need that policy to stay ahead. The one-time boost can be useful, but it's better if you can slot their GWoW into a building that will give you a bonus to your Tourism.