What are the most commonly used wave bands for communications between satellites in LEO and ground station, and what is the minimum frequency used?
I'm going to address this question as per the US FCC, because it seems similar rules apply anywhere. The FCC has a chart which shows the various frequency allocations. For the first question, the lowest frequency signals appear to be Amateur Satellites, namely, 29300-29510 kHz. For commercial/government satellites, they seem to be limited to the VHF range, I see some in the 146-138 range, for instance, and that seems to be the highest supported.
So, given that, why would there not be a lower frequency? And why don't Amateurs seem to care? Low frequency requires larger antennas, usually shaped in a Yagi configuration. Higher frequencies can use dish type antennas, and typically get higher gain than they would otherwise. Also, lower frequency= lower bandwidth, thus, they can't transmit data very fast. Finally, they tend to bounce around the Ionosphere, not always leaving the atmosphere, and can be confused with signals very far apart. Furthermore, they are noisier, the Sun puts out a lot of energy at lower frequencies. As a whole, they are just clunky for orbital operations. Amateur Radio Operators don't typically care, as they often have large antennas, and only want to do simple communication with others (Voice, small data, Morse Code, etc)
As to what frequency is most commonly used, one of the goals to planning satellites is to get them to not use exactly the same frequency, so that they don't interfere with each other. Not withstanding, the most common band for communication satellites is the KU Band.
The Model-1 (1980) and the early Model-2 (1982, 1983) experiments have failed to deploy properly the elastic antennas.
The last of the Model-2 (1987, with an inflatable antenna) and Krab (1989, shape-memory alloys) have been successfuly deployed and several communication sessions between the ground station and Progress spacecraft have been established.
See this video @ 32:00 for an unsuccessful and a successful Model deployments.