Just writing up this quick set of guidelines to aid people (including myself) in writing good solutions and proofs.
1) Keep your audience in mind. Show every essential step of your working, so that the reader isn't puzzled by your solution and doesn't have to guess at your motivations; at the same time, don't show every single step, as it may come off as condescending. A good balance makes a great solution.
2) Don't write all of it in equations. Include prose which outlines what theorems/lemmas/results you are invoking. A solution chock full of line after line of equations is hardly a pretty sight. Keep it clean.
3) Still, don't go rambling on. It's very annoying. Leave that for the comments section, where you can append a footnote or two.
4) Use LaTeX. It makes everything easier to read and understand, and clarity is paramount.
5) But don't wrap everything in LaTeX! It's extremely difficult to read. I've seen this a lot on Brilliant. Far too much, actually. It lessens your load, too, so hey! Win-win for both the writer and the reader.
6) Use connectors such as "therefore", "hence", "since", "because", etc to keep the solution cohesive as a whole. They provide a good sense of flow within any proof, which makes it easier for the reader to process.
Keep a good balance of \ [ \ ] and \ ( \ ). As a rule of thumb, a good ratio lies between 2:1 and 1:2. This helps to keep yourself in check, to avoid verbosity by including only important equations/expressions/etc worth wrapping in \ [ \ ].
and other special functions are given to you, so use them. I don't know about others, but it's not pleasant to see a stray in a solution, personally.
Provide useful links to theorems and results. Good websites for reference include MathWorld and Wikipedia.
Don't squeeze a fraction into a tiny space, eg . It doesn't look good, and it's hard to read. Try or instead!
Use lemmas sparingly. Easier questions (mostly) don't require lemmas, but it will be apparent to the advanced problem solver when (and when not) to use lemmas.
I hope that at least somebody out there will be inspired to increase the quality of their proof-writing! I strongly feel that lucidity in writing is always quintessential for any sort of mathematical writing, and so these guidelines serve to help one improve upon that important aspect.
Anything you would like to add? Comment and I will add on to these guidelines.