endian-ness of bits and bytes

Big Endian, Little Endian... What it means for bytes is well known - in a multi-byte word (for example, the binary representation of an integer on a 32 bit machine), little-endian systems (Intel) will have the byte with the lsb in the lowest address, big-endian systems (IBM/Motorola, etc) will have it in the highest address.

But what does endian-ness mean on bit level ? To rephrase, say I have a file... I write one byte to it, the character 'A'. It's 65, so "01000001" in binary. In that file, what bit will be "the first bit of the file" ? 1 or 0 ?

It seems that this is not defined anywhere, because bytes are the lowest people go on PCs. But sometimes, an answer for this question is needed, say if the byte was received from a serial bit-stream...