As far as I know, there are five different solving instruments, namely using an applet (online or offline), an image editor program, a spreadsheet program, a whiteboard/blackboard, and hand-written. I will try to give my opinion on each. Remember, this is just an opinion, and the methods are not listed in any specific order (though it seems like being listed in the best/easiest to use to the worst/hardest to use, it might not).
Perhaps the best solving instrument. An applet can save you a lot of time of filling unnecessary cells (see Akari), and can also reminds you if anything is wrong. However, applets are not foolproof. There must be someone that wants to make the applet for each puzzle type, and you need a computer access (or an electronic device capable of using the applet) to use the applet. Almost, if not all, puzzles can be aided well with applets…if the applet for the puzzle type is available. Akari definitely goes here; none of the other three instruments can aid an Akari solving at least close to how an applet aids an Akari solving.
2. MS Paint or any image editor program
A good alternate of applets that is widely available. However, this can’t give good enough speed in quick solving, as you might need to change between colors or tools often. Image editors are also incapable of “trying” a path, since undoing can take quite a lot of time. The puzzle types that I can think at the time and can be aided well with image editors are Nurikabe, Minesweeper, Tapa, Hitori, etc; basically those that only needs to shade cells, and only needs two colors.
3. MS Excel or any spreadsheet program
Also what I currently use to post new puzzles, a spreadsheet program is great to type in letters and numbers, IF AND ONLY IF none of the givens or the symbols have weird borders (see Irregular Sudoku and Fillomino). Colors? Completely available; 16777216 colors to pick from (though some are strikingly similar, see #FFFFFF (white) and #FEFEFE (off-white)). Even using only web-safe colors that are dark enough gives about 60 (216 divided by around 3-4) colors to pick from, more than enough to branch out even ten possibilities. The puzzle types that I can think at the time and can be aided well with a spreadsheet program are Sudoku, Kakuro (except the puzzle copying), Skyscraper, etc; basically anything that requires filling letters and numbers and only filling letters and numbers.
4. A whiteboard or blackboard
Yeah, I do think of this one. The various colors of board markers and chalks gives these instruments a capability of trying a path. Moreover, this variety of colors also allows solving puzzles that are multi-colored (MellowMelon’s Colorlink for example). The puzzle types that I can think at the time and can be aided well with a whiteboard or a blackboard (and not hand-written) are…uh, well, probably only Colorlink that I can think at the time.
5. Hand-written (with pencil; pen is definitely worse except for the colors, and colored pencils work well)
The old way of solving a puzzle. Either print out the puzzle, which requires additional paper and printer, or copy the puzzle manually, which requires a reusable (and not messy) graph paper and quite some effort for gigantic puzzles. There are quite a lot of downsides to hand-written; say, can’t try a path without some colored pencils, you might mix up which ones are the givens (if you copy manually), etc…but you can use just all notations and symbols and shapes you can think of for the puzzle, and you can write letters or numbers well (try writing a digit by brush in an image editor; you’ll notice how weird it is). The puzzle types that I can think at the time and can be aided well hand-written are Fillomino, Battleships, Corral, Slitherlink, etc; basically those that requires writing letters/numbers/symbols/borders.
P.S. Probably my first use of the “More tag”. Hey, it works neatly!