Which use of practised is correct, if any?
I practised my singing.
I practised at singing.
I practised singing.
The 'my' is not technically incorrect, as far as I know, but is superfluous. It would be difficult to practice someone else's singing :)
One can become practiced at something, as a passive verb, but to practice at something directly is not grammatically correct.
This is the preferred usage.
I would use this if it's clear from the context what exactly you were singing. For instance a piece my teacher assigned to me or something I sing on a regular base.
This sounds wrong to me. "At" should be used for the location you were practising at. I practised singing at school.
That's how I usually use it.
The are two words for the word practice.
Below are correct examples of how to use the two terms.
In one of your three examples, you used the word incorrectly.
The usage of the word at, makes the word singing feel like a noun, which it isn't.
How is it possible to practise at singing? How is it possible to trip at walking? You can trip your walking? You can trip walking? But you cannot trip at walking!
The word at is a preposition, so it shouldn't be used in that manner, because prepositions are used for nouns.