Stardock Weather Icons
Set of 49 icons - PNG files - 128×128

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Flat Weather Icons
Set of 105 icons - PNG files - 128×128 - Also comes in different color and language variations

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Iconbest Weather Icons
Set of 49 icons - PNG files - 128×128

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Shiny Weather Icons
Set of 105 icons - PNG files - 256×256

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Oslo Atmosphere Weather Icons
Set of 9 icons - Multiple filetypes - 16×16 to 32×32

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Alexandria Atmosphere Weather Icons
Set of 9 icons - Multiple filetypes - 16×16 to 32×32 - Multiple color variations

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Manhattan Atmosphere Weather Icons
Set of 8 icons - Multiple filetypes - 16×16 to 32×32

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Not Quite So Free (But still inexpensive)

Worth noting is that all of these are vector icon sets, making them significantly more flexible than the free sets mentioned above.