  We study the two-dimensional stochastic nonlinear wave equation (SNLW) and
stochastic nonlinear heat equation (SNLH) with a quadratic nonlinearity, forced
by a fractional derivative (of order $\alpha > 0$) of a space-time white noise.
In particular, we show that the well-posedness theory breaks at $\alpha = \frac
12$ for SNLW and at $\alpha = 1$ for SNLH. This provides a first example
showing that SNLW behaves less favorably than SNLH. (i) As for SNLW, Deya
(2020) essentially proved its local well-posedness for $0 < \alpha < \frac 12$.
We first revisit this argument and establish multilinear smoothing of order
$\frac 14$ on the second order stochastic term in the spirit of a recent work
by Gubinelli, Koch, and Oh (2018). This allows us to simplify the local
well-posedness argument for some range of $\alpha$. On the other hand, when
$\alpha \geq \frac 12$, we show that SNLW is ill-posed in the sense that the
second order stochastic term is not a continuous function of time with values
in spatial distributions. This shows that a standard method such as the Da
Prato-Debussche trick or its variant, based on a higher order expansion, breaks
down for $\alpha \ge \frac 12$. (ii) As for SNLH, we establish analogous
results with a threshold given by $\alpha = 1$.
  These examples show that in the case of rough noises, the existing
well-posedness theory for singular stochastic PDEs breaks down before reaching
the critical values ($\alpha = \frac 34$ in the wave case and $\alpha = 2$ in
the heat case) predicted by the scaling analysis (due to Deng, Nahmod, and Yue
(2019) in the wave case and due to Hairer (2014) in the heat case).
