In this thesis we study homological stability properties of different families of spaces using the technique of cellular E_k-algebras. Firstly, we will consider spin mapping class groups of surfaces, and their algebraic analogue —quadratic symplectic groups— using cellular E₂-algebras. We will obtain improvements in their stability results, which for the spin mapping class groups we will show to be optimal away from the prime 2. We will also prove that in both cases the $F$₂-homology satisfies secondary homological stability. Finally, we will give full descriptions of the first homology groups of the spin mapping class groups and of the quadratic symplectic groups. Secondly, we will study the classifying spaces of the diffeomorphism groups of the manifolds W_g,1 ∶= D²ⁿ#(Sⁿ x Sⁿ)^#g. We will get new improvements in the stability results, especially when working with rational coefficients. Moreover, we will prove a new type of stability result —quantised homological stability— which says that either the best integral stability result is a linear bound of slope 1/2 or the stability is at least as good as a line of slope 2/3.