In the topological category the usual compact-open topology does the job. At times you might want to replace a function space by the simplicial set called its total singular complex. This is an instance of the more general procedure of replacing a space by its total singular complex when convenient, a trend that began around that same time. But there may not be any great technical advantages in doing so when working with spaces of homeomorphisms.
In the smooth category you can give the space of diffeomorphisms the Whitney topology and then look at the total singular complex, but in some ways it's a better idea to look at the smaller simplicial set consisting of the "smooth" (rather than merely continuous) maps of simplices into the space, meaning those which correspond to smooth maps $\Delta\times M\to M$. Note that when dealing with smooth bundles with fiber $M$ over a base $B$ you really are interested more in "smooth" maps from open subsets $U\subset B$ into the group than in continuous maps, because it is the former that correspond to automorphisms of the trivial bundle $U\times M$. It is true that, just as for maps between finite-dimensional manifolds, smooth homotopy classes of smooth maps from $U$ to the group are in bijection with homotopy classes of continuous maps from $U$ to the group. It is not true that the compact-open topology on smooth automorphisms is just as good as the Whitney topology for these purposes: you don't want to be able to kill an exotic diffeomorphism of $D^n$ that fixes a neighborhood of the boundary sphere by the Alexander trick (which scales it through diffeomorphisms until at the last moment it becomes the identity).
In the $PL$ category it gets worse: I don't know a useful topology on the group of $PL$ automorphisms of $M$. Topologizing it as a subspace of the space of homeomorphisms can't be right, and there isn't something like a Whitney topology to fix things up.
One other point:
What about the structure group for microbundles? You want to compare on the one hand the group of automorphisms of $\mathbb R^n$ fixing the origin (the structure group for open disk bundles with section) and on the other hand the group of invertible germs of maps $\mathbb R^n\to \mathbb R^n$ at the origin (the structure group for microbundles). As I understand the Kister-Mazur Theorem, bundles are "the same" as microbundles because the comparison map of groups is an equivalence. The kernel of the comparison map (i.e. the group of all automorphisms of $\mathbb R^n$ such that some neighborhood of the origin is pointwise fixed) is rather obviously contractible, while on the other hand the map is surjective (every invertible germ is the germ of a global automorphism). I think that in getting the details of that right it is convenient not to have to actually topologize the group of germs, but rather to speak of germs of maps from $\Delta^k\times \mathbb R^n$ to itself along $\Delta^k\times 0$.