Waves on the sea or in the clouds are some of the most simple, yet beautiful, natural dynamic phenomena on Earth. They sometimes roll-up and form growing whirls. These phenomena occur due to a large flow difference at the transition between the air and water or cloud, which makes the boundary unstable. Similar instabilities also take place in space, when the filling plasma, a mixture of charged particles, is exposed to various sources of flows. The solar wind, for instance, is a continuous stream of plasma flowing away from the Sun. The magnetic field of the Earth forms a protective bubble called the magnetosphere, which is affected by large disturbances in the solar wind. The boundary between the solar wind and the magnetosphere is notably subject to large boundary motions. These motions may be the response to the changing force of the solar wind, like a fluttering windsock. They may also be growing whirls, that could involve a mixture of the two plasmas. In the solar wind, one can find massive clouds of material, called Coronal Mass Ejections (CMEs). CMEs are ejected from the solar atmosphere at hundreds of km per second, and are associated with outflows, another source of abrupt flow changes and instabilities. Very similar outflows are generated inside the magnetosphere. Space observations are challenging to analyse, the complexity of plasma physics makes them interesting to study and understand. I have long been fascinated by the natural world and its impact on us. The Sun affects life on Earth in many ways. Dangerous, high energy, charged particles are often produced in the CME disturbances. The magnetosphere shields us from most harmful solar effects. However, the large disturbances in the solar wind affect our climate and causes, for instance, communication drop-outs, power outages and radiation exposure on transatlantic flights. In the Sun's outer atmosphere, the solar corona, there are relatively cool, dense clouds of gas, which we call the solar prominences (or filaments). Prominences are in constant motion and display a relatively fast dynamic and turbulent flow system. Sometimes prominences erupt off the Sun, along with CMEs. From space, we can observe prominences night and day without interruption. Doing so, I have discovered prominences which oscillate quite slowly, so slowly that it is almost impossible to detect it continuously from Earth. The oscillations seem connected to prominence eruptions. Is there a link between the two types of dynamics, fast and slow? Is this link relevant to other flow instabilities in space? How are the prominences formed and why do they erupt? How is the solar wind accelerated? How does the solar wind get through the apparently closed field lines of the Earth's magnetic field? To find out, I propose to get a better picture of the flow instabilities in regions of the Sun-Earth system, where they seem to play a major role. All these phenomena are observed with a telescope or with a satellite measuring the plasma properties in space. Such observations present complementary perspectives. Their ever-increasing number and quality (e.g. Solar Dynamics Observatory) make it the right time to compare the phenomena, find their common features and their differences. I find it both exciting to develop a unifying view (combining my past experiences in both research fields) and important for our society to shed light on the processes at work in key regions of the Sun-Earth system.