  We consider singular quasilinear stochastic partial differential equations
(SPDEs) studied in \cite{FHSX}, which are defined in paracontrolled sense. The
main aim of the present article is to establish the global-in-time solvability
for a particular class of SPDEs with origin in particle systems and, under a
certain additional condition on the noise, prove the convergence of the
solutions to stationary solutions as $t\to\infty$. We apply the method of
energy inequality and Poincar\'e inequality. It is essential that the
Poincar\'e constant can be taken uniformly in an approximating sequence of the
noise. We also use the continuity of the solutions in the enhanced noise,
initial values and coefficients of the equation, which we prove in this article
for general SPDEs discussed in \cite{FHSX} except that in the enhanced noise.
Moreover, we apply the initial layer property of improving regularity of the
solutions in a short time.
