I have seen many ground state wave functions.
From where are they derived? How can one calculate them? Where can one find a list of all ground state wavefunctions discovered?
I'm not sure on what level you're asking this, so ignore the bits of this answer that don't make sense.
Wavefunctions, ground state and otherwise, aren't designed. For some system, e.g. a hydrogen atom, you write down the Schrodinger equation and solve it. The solutions are the wavefunctions. All you need to know to write down the Schrodinger equation is the potential energy of your system e.g. for the hydrogen atom this is the electrostatic attraction between the electron and proton.
However life is rarely this simple. For example the Schrodinger equation is only an approximation and doesn't take relativistic effects into account. For that you need the Dirac equation and the extra complexity of this makes exact wavefunctions impossible to write down. Instead we start with the solutions of the Schrodinger equation and include the relativistic effects as perturbations.
Even sticking to the Schrodinger equation, all but simple systems are impossible to solve exactly and we have to use approximate methods and wheel out our computers. Even the H$_2$ molecule doesn't have a simple exact solution.
There is no list of all ground state wavefunctions discovered. Any such list would be enormously long. Did you have any particular examples in mind?