As a quick example, Javascript is not designed to handle leap seconds correctly, so I doubt we'll ever see the value 60 out of new Date().getSeconds(). Things will just be off by 1 second for a second.
Edit: Another example from the Windows Time service:
> The Windows Time service does not indicate the value of the Leap Indicator when the Windows Time service receives a packet that includes a leap second. [...] Therefore, after the leap second occurs, the NTP client that is running Windows Time service is one second faster than the actual time. This time difference is resolved at the next time synchronization.
It will just be a second off and resynchronize next time.