It is rather dependent on the surveying method used! If you are using the RTK which is the most popular today, your accuracy level is at cm level. If you are using rapid static GPS surveying along with rapid static processing software such as OPUS-RS or BERNESE, your accuracy is again at cm level. Examples show that with careful experiment design and processing strategies it is possible to improve the accuracy of rapid static positioning from cm to mm! However this needs a lot of effort and knowledge. On the other hand, if you are using static GPS then you can obtain mm level. The best option is to set up GPS antennas on concrete pillars and collect 24h data. Afterwards you need to apply proper time-series analysis to gain mm deformation rates. If your observing session duration is less than 24 h then the accuracy becomes worse as low as 5 mm for horizontal coordinates and 2 cm for the vertical component (with 2 h of data!). The above figures from PPP processing. With relative positioning over short baselines (say 10-20 km) you can obtain much better precision with 2 h data. Hope this helps.
It depends on series of circumstances, e.g. the satellite system (Navstar GPS, GLONASS, etc.), the GPS receiver, the method used to observe satellites, the time of observation, data processing and adjustment method. Some mm-s of positioning accuracy can be expected, with 1,5 - 2 times decreasing in vertical component.
I agree with the above researchers. The positioning accuracy to be achieved is heavily dependent on the positioning method you are going to use (e.g. RTK, PPP), the time interval of your observations, the satellite geometry at the time of your observations, the environment of receiver (be aware of multipath in urban canyons) and the GNSS signals you are tracking. The last depends on the GNSS receiver you use, and just for your information recent studies show that using multiple GNSS constellations (so your GNSS receiver should be able to track signals not only from GPS and GLONASS) can enhance the position accuracy achieved as well as the convergence time.