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I'm having some trouble finding what `フッたん` means, I can see that `降った` is
read the same, but I've found is never even rendered in Hiragana let alone
this way. I'm not even particularly sure if `フッたん` is something by itself.

The full piece of dialogue is the following:

> サイッテ~ ...彼氏と別れた~。
>
> ...私がフッたんだけどさ~...。~

A bit more context is that it is a girl speaking to two others.
CC BY-SA 3.0
2016-01-20T08:04:48.763
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2016-05-20T23:08:40.030
2020-06-17T08:18:27.500
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meaning
Meaning of フッたん
フッたんだ is a colloquial way of saying ふったのだ.

  * ふる: to ditch (someone), to dump (someone). See definition 5 in [this entry](http://dictionary.goo.ne.jp/jn/195909/meaning/m0u/). It's 振る in kanji, but in this sense, it's usually written in hiragana and [sometimes in katakana](https://japanese.stackexchange.com/q/1930/5010).
  * ふっ + た: te-form of ふる + た denoting past
  * んだ = のだ. See [this](https://japanese.stackexchange.com/q/23936/5010).

So 私がフッたんだけどさ means "It's me who dumped him, though."
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2016-01-20T08:16:13.430
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2016-01-20T08:22:03.433
2017-04-13T12:43:44.397
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answer
The whole dialogue is very colloquial, and it's quite common when rendering
colloquial speech in writing, especially manga, to use katakana to render
emphasis.

You did not make any remark about it, but the first word of the dialogue is a
very typical example of this, so is a big hint when we see the フッたん later that
it's still the same thing happening :  
サイッテー which would normally be 最低{さいてい} ("the worst", "what a shit") shows
three colloquial effects: use of katakana instead of kanji/hiragana, えい at the
end changing into an lengthened え, and an extra っ before the て to mark a
chopped style of speaking for extra emphasis.
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2016-01-20T10:10:27.220
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