  We generalise a technique of Bhat and Skeide (2015) to interpolate commuting
families $\{S_{i}\}_{i \in \mathcal{I}}$ of contractions on a Hilbert space
$\mathcal{H}$, to commuting families $\{T_{i}\}_{i \in \mathcal{I}}$ of
contractive $\mathcal{C}_{0}$-semigroups on $L^{2}(\prod_{i \in
\mathcal{I}}\mathbb{T}) \otimes \mathcal{H}$. As an excursus, we provide
applications of the interpolations to time-discretisation and the embedding
problem. Applied to Parrott's construction (1970), we then demonstrate for $d
\in \mathbb{N}$ with $d \geq 3$ the existence of commuting families
$\{T_{i}\}_{i=1}^{d}$ of contractive $\mathcal{C}_{0}$-semigroups which admit
no simultaneous unitary dilation. As an application of these counter-examples,
we obtain the residuality wrt. the topology of uniform wot-convergence on
compact subsets of $\mathbb{R}_{\geq 0}^{d}$ of non-unitarily dilatable and
non-unitarily approximable $d$-parameter contractive
$\mathcal{C}_{0}$-semigroups on separable infinite-dimensional Hilbert spaces
for each $d \geq 3$. Similar results are also developed for $d$-tuples of
commuting contractions. And by building on the counter-examples of
Varopoulos--Kaijser (1973--74), a 0--1-result is obtained for the von Neumann
inequality. Finally, we discuss applications to rigidity as well as the
embedding problem, \textit{viz.} that `typical' pairs of commuting operators
can be simultaneously embedded into commuting pairs of
$\mathcal{C}_{0}$-semigroups, which extends results of Eisner (2009--10).
