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In this example, Einojuhani Rautavaara sometimes uses flat and sometimes uses sharp in some chords.

Once it's atonal music, what're the rules for sharps and flats?

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    As far as I can tell, each composer uses his/her own rules.
    – Your Uncle Bob
    Aug 13, 2019 at 0:29
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    Not true, uncle bob.
    – jjmusicnotes
    Aug 13, 2019 at 2:26
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    It's also worth thinking about whether or not this is 'truly atonal'. It doesn't sound like twelve-tone serialism. While it's not centred on a standard key it is relatively consonant. It's using chords which will tend towards a tonal centre or implied root note. It's possible the sharp/flat decision is based on 'spelling' the most coherent chords.
    – AJFaraday
    Aug 13, 2019 at 9:43
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    @jjmusicnotes If the rules you put forward are indeed a kind of standard, and not just common sense and personal preference, then a few examples would give your answer more weight.
    – Your Uncle Bob
    Aug 13, 2019 at 13:23
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    The "rules" for the strict note-naming so far as theories of harmony go are one thing. The rules for marking the parts to be played are laid out pretty well in jjmusicnotes' answer.
    – Carl Witthoft
    Aug 13, 2019 at 14:22

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Fair question. Think about these things:

1.) Context (lots of sharps / flats already? Which would simplify the music?)

2.) The direction (sharps = up / flats = down)

3.) The instrument (strings more comfortable reading sharps / winds more comfortable with flats)

4.) Writing intervals the way they sound (does that really need to be "C-D#" or would "C-Eb" make more sense?)

5.) Voice-leading (this is part of "context" from above - what makes the most sense voice-leading wise? Atonality and functional harmony can coexist; just because your chords have functional voice-leading does not mean that they can't exist within an atonal framework.)

Make the decisions that make the most sense. The more you can simplify the music, the better.

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