Batteries can 'go' bad/dead/flat, but since you're describing them when they're already in that state, I'd say 'are bad/dead/flat', or 'have gone bad/dead/flat'.
To me, 'dead' and 'flat' are the same. However, it just means they don't have any power left in them. They aren't corroded or leaking.
Sometimes we do refer to the corroded/leaking ones as 'bad', but I'd never use 'dead' or 'flat' to describe leaky ones. I'd either say 'bad' or 'corroded', or maybe 'leaking.'
However, 'bad' can be used to mean the same as 'dead' and 'flat' as well.
They're probably all equally as common. I alternate between all three terms myself.
Also - you leave batteries 'in' a device, not 'on' them.
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