Let $X_{k,n}$ be the space consisting of $k$ copies of $\mathbb{R}^n$ all identified at a single point. For example, $X_{1,n} = \mathbb{R}^n$ and $X_{2,2}$ is two planes pinched together at a point, i.e. a 'conic section' cone.
You could try to define a "$(k,n)$-cone manifold" as a paracompact $T_2$ space each of whose points has a neighborhood homeomorphic to $\mathbb{R}^n$ or some $X_{k,n}$ for fixed $n$ - alternatively, to $X_{l,n}$ for $l \leq k$ for $k \in \mathbb{N} \cup \lbrace \infty \rbrace$. Do such spaces have a name, and are there any general theorems about them? Mostly their algebraic properties I assume are already completely worked out.
In this case, the pinched points will be discrete, which is kind of boring. So instead recall the definition of the Cantor-Bendixon derivative $\alpha$ of a topological space $X$. Namely, it is $\alpha(X) = X \setminus \text{(isolated points of } X)$. The order type of a (non-empty) countable set is the minimal number of times you have to take the derivative to get the empty set - if none exists, we say its order type is infinite. So the order type of a (non-empty) discrete set is $1$, and otherwise it's higher.
Let $\mathcal{D}_m^n$ be the collection of closed, countable subsets of $\mathbb{R}^n$ with order type $m$ (alt. less than or equal to $m$), where $m$ is finite. We could define an $(n, k, m)$-cone manifold to be a paracompact Hausdorff space such that every point has a neighborhood homeomorphic to $\mathbb{R}^n$ or a quotient of $k$ (alt. less than or equal to $k$) copies of $\mathbb{R}^n$ along any $D \in \mathcal{D}_m^n$. This seems like the next-nicest generalization.
But this construction doesn't allow "pinches between the pinches." How would you define a fourth parameter, something like a 'magnitude' or 'depth,' that allows a neighborhood $U$ of a point to be something like "any point in $U$ is from an $(n, k, m)$-cone manifold" without allowing the space to be torn up?