How do you increase a number *directly* under the cursor?

In this case, I usually type r and then the digit I want.

<C-a> and <C-x> work on whole numbers (1 or more digits) but not on "parts" of a number. However it takes a "count" so 10<C-a> turns [6]5 into 75.

Another option, since you only want to change one digit, would be to do r7 to turn [6]5 into 75.

As I guess 3[9]42 shall become 4042 (otherwise, Xavier's solution is almost perfect (*)), I'd use the following:

s/\d*\%#\d/\=(submatch(0)+1)

Which could be mapped into:

nnoremap <silent> µ :<c-u>s/\d*\%#\d/\=(submatch(0)+v:count1)<cr>``

in order to accept a count. The only catch I see is that it cannot support repeat (as in :h .)

(*) In the sense that repeating the command won't help us to increment a series of different numbers by the same amount.