Institut d'Astrophysique Spatiale (France): There are a few opportunities each year to work within the solar physics team at IAS, from Bachelor's and Master's level traineeship to getting enrolled in a PhD program. For prospective candidates to a PhD program, there is a national funding available every year with a competitive scheme.
These routes on hold for the time being while I am seconded to the European Space Agency.
European Space Agency (Netherlands): A few opportunities exist to work within the division of science and in the field of solar and space physics.
Young Graduate Trainees: see info here
The Leiden/ESA Astrophysics Program for Summer Students: see info here: