Gumshoe Online is a single-player online detective game created by Hiding Buffalo LTD. There are currently four cases available, with the first one, 'The Osbourne Mystery', being free to play while the others have a very minimal charge.
Note: registration is required to play even the free trial mystery.
It should be pointed out that the trial mystery is very limited and that the actual cases have a much greater length and depth. My favourite case, with respect to its story, is 'Higher Order of the Three', but if you wish to purchase only one then I'd highly recommend the most recent one - 'Something in the Water' as it's easily the most enjoyable.
Each case is made up of a number of detective-like elements. For example, you will spend the majority of your time searching around various locations looking for clues. Each location has dozens of items that can be examined and most rooms have at least one minor clue. Another fairly small section is interviewing people. This is very limiting, as most individuals have only a handful of topics that they can be questioned on, and the tolerance system is fairly useless.
Finally are the puzzles. These are the most enjoyable and at the same time infuriating parts of the game. There are a fairly wide variety of puzzles, from assembling a fuse box to testing a water sample. Some of them rely on math, while others are simply logic. Many of the puzzles are fairly difficult, while some of them are almost impossible, as can be seen by the extremely popular forums, the best place to ask for help with the game.
When you've gathered all of the evidence that you feel you need, you can go back to your office and come up with a solution. If you have the right solution this part is fairly trivial, but it can be annoying when you're positive about your solution and it turns our to be incorrect, as no help is given to push you in the right direction. (A similar system may also help in lowering the difficulty of some puzzles). It's vital that you save before submitting a solution, as if it's wrong you'll have to do the whole case again or just reload your save. After submitting a correct solution, you get a score based on the time (in-game) that it took you and how many clues you collected.
Despite my criticisms, I really enjoyed Gumshoe Online and felt that all of the cases were well worth the cost - I fully recommend buying at least one. Click.
Cheers to Luckdragon for suggesting the game. =)