  Shape and multiple shape coexistence of nuclei are investigated throughout
the nuclear chart by calculating the low-lying spectra and the quadrupole shape
invariants for even-even nuclei with $10\leq Z\leq 104$ from the proton drip
line to the neutron one within a five-dimensional collective Hamiltonian based
on the covariant density functional PC-PK1. The quadrupole shape invariants are
implemented to characterize the quadrupole deformations of low-lying $0^+$
states and predict nuclear mass regions of shape and multiple shape
coexistence. The predicted low-lying spectra and the shape or multi-shape
coexisting nuclei are overall in good agreement with the available experimental
results. In addition, the present work predicts a wealth of nuclei with shape
or multiple shape coexistence in the neutron-rich regions. The connection
between the strong $E0$ transition strength and the occurrence of shape
coexistence is analyzed systemically. It is found that nuclei with pronounced
shape coexistence generally have strong $E0$ transition strengths, while the
reverse may not be true. The present results can serve as useful guidelines for
experimental searches and theoretical studies of shape and multiple shape
coexistence, especially in neutron-rich regions.
