I have gotten the "see vote counts" privilege on Stack Overflow, and I visited a lot of downvoted and upvoted posts on Stack Overflow on the mobile site, but I can't see number of upvotes and downvotes.
Does the mobile site not support it?
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No, the mobile site does not support it. Early versions of the app did support it (might have just been development versions), but a lot of phones at the time didn't provide enough accuracy on the touchscreen to make it useful (I asked this same question several years ago).
Perhaps now that a lot of phones have larger screens, this feature may be more feasible...