When I use bench power supply such as below:
I use the red terminals as (+) and black terminals as (-) and also as ground. But the green terminal I never use and it is connected to mains earth ground.
The PSU as a whole is already earth grounded from its 3-prong AC plug. I couldn't figure out a use case for the green terminal. Why/when would one want to use the earth ground as his circuit ground? A few examples would help.
The black and red outputs are isolated and floating. They have no relation to earth or mains input.
You use the green earth pin if you want to reference some point of your device to earth. If not then don't use it.
A case when you might want to use the green earth terminal is if you have a device with metal case and you want to make sure the metal shield is not floating but connected to earth. The device might be a sensitive amplifier or a high voltage generator. When earthed, it can reduce surrounding noise being coupled to circuit, or it may add safety if your high voltage generator malfunctions and the case comes live with high voltage.
And there are also cases when you likely don't want to use it. For example if your powered device also connects to another earthed device (like desktop PC over USB or serial connection), if the ground between device and PC malfunctions, the return current will flow via mains earth wiring inside walls.
I own a linear lab psu made in 1971 with a 723 reg IC and a 2n3055 pass transistor.I would notice a 100Hz buzzing sound when using it on AM car radios .Earthing the black negative lead to the case would cure the noise .Using a 1microfarad film cap between the neg lead and the case also cures the buzzing noise. Now some cars and hence car radios of yesteryear are positive earth and many pnp germanium equipment is also positive earth. Later model lab power supplies have the green case earth terminal to make this more conveniant. When there is lots of plastic like the one in your photo it makes the point to connest the earth clear .
