The examples considered in algebraic topology are usually like $\mathbb{R}^\infty=\lim_n\mathbb{R}^n$ or $O(\infty)=\lim_n O(n)$ (the infinite orthogonal group) or infinite-dimensional CW complexes. (Here $\lim$ indicates a colimit; I would write that more explicitly but something funny is happening with MathJax at the moment.) These can usually be written as the colimit of locally compact Hausdorff spaces $X_n$ and closed inclusions $X_n\to X_{n+1}$, where $X_n$ typically has empty interior in $X_{n+1}$. In this context, a standard lemma says that every compact subset $K\subseteq X$ is contained in some $X_n$ and so has empty interior in $X_{n+1}$ (and therefore also has empty interior in $X$). Thus, examples of this type will never be locally compact.
One could consider the class $\mathcal{C}$ of $k$-spaces in which every compact set has empty interior, which is a kind of opposite to local compactness. I think that this class of spaces contains most of the popular examples and has good closure properties.
On the other hand, very many spaces can be written as a quotient of an LCS space. Indeed, let $CS(X)$ be the set of continuous maps $\mathbb{N}\cup\{\infty\}\to X$ (i.e. pairs consisting of a convergent sequence and its limit). We can give $CS(X)$ the discrete topology and the space $CSN(X)=CS(X)\times(\mathbb{N}\cup\{\infty\})$ the product topology. This makes $CSN(X)$ into an LCS space, with an evident surjective evaluation map $\epsilon\colon CSN(X)\to X$. The space $X$ is said to be sequential if $\epsilon$ is a quotient map. (The definition is usually phrased differently, but easily seen to be equivalent.) One can check that if $X$ is a $k$-space and every compact subspace is metrisable then $X$ is sequential. This covers most examples typically considered in algebraic topology.
My guess is that the space $\beta(\mathbb{N})$ (the Stone-Čech compactification of the naturals) is not a quotient of an LCS space, but I have not checked that.