  The study of magnetohydrodynamics (MHD) significantly boosts the
understanding and development of solar physics, planetary dynamics and
controlled nuclear fusion. Dynamical properties of the MHD system involve
nonlinear interactions of waves with multiple travelling speeds (the fast and
slow magnetosonic waves, the Alfv\'{e}n wave and the entropy wave). One
intriguing topic is the shock phenomena accompanied by the magnetic field,
which have been affirmed by astronomical observations. However, permitting the
residence of all above multi-speed waves, mathematically, whether one can prove
shock formation for 3D MHD is still open. The multiple-speed nature of the MHD
system makes it fascinating and challenging. In this paper, we give an
affirmative answer to the above question. For 3D ideal compressible MHD, we
construct examples of shock formation allowing the presence of all
characteristic waves with multiple wave speeds. Building on our construction,
we further prove that the Cauchy problem for 3D ideal MHD is $H^2$ ill-posed.
And this is caused by the instantaneous shock formation. In particular, when
the magnetic field is absent, we also provide a desired low-regularity
ill-posedness result for the 3D compressible Euler equations, and it is sharp
with respect to the regularity of the fluid velocity. Our proof for 3D MHD is
based on a coalition of a carefully designed algebraic approach and a geometric
approach. To trace the nonlinear interactions of various waves, we
algebraically decompose the 3D ideal MHD equations into a $7\times 7$
non-strictly hyperbolic system. Via detailed calculations, we reveal its hidden
subtle structures. With them we give a complete description of MHD dynamics up
to the earliest singular event, when a shock forms.
