Updated: Jan 31, 2021
SpaceX was finally able to hop their Starship prototype SN5 (serial number 5) yesterday, August 4th to 150 meters. Check out the video from SpaceX below. Note that this is just the bottom half of the upper stage with only one engine. Instead of a block of nearly solid steel at the top like this prototype had, the complete upper stage will be about twice as tall with the top half being the cargo area or crew area depending on the configuration. Everything would in turn be placed on top of the huge super-heavy booster currently under development.
SpaceX plans to do several more short hops before a much higher one with body flaps, though with SN6 ready to go already and SN7 and SN8 under construction, it is unclear which one will do the higher altitude flight.
According to Elon Musk, upcoming versions will also have larger landing legs that can handle uneven terrain (as would be needed for a moon or Mars landing).