We give a survey of techniques from quantum group theory which can be used to
show that some quantum spaces (objects of the category dual to the category of
$\mathrm{C}^*$-algebras) do not admit any quantum group structure. We also
provide a number of examples which include some very well known quantum spaces.
Our tools include several purely quantum group theoretical results as well as
study of existence of characters and traces on $\mathrm{C}^*$-algebras
describing the considered quantum spaces as well as properties such as
nuclearity.