Background
The $\mathcal{H}$-factor problem (a.k.a. the degree prescribed factor problem, or the degree prescribed subgraph problem) is defined as follows:
Given a graph $G=(V,E)$ and a set $H_v \subseteq \mathbb{N}$ for each vertex $v \in V$, does $G$ contain a spanning subgraph $F$ such that $\operatorname{deg}_F(v) \in H_v$ for all $v \in V$?
(I would also say that $H_v \in \mathcal{H}$ for all $v \in V$, but I have never seen it stated this way. Thus, the $\mathcal{H}$-factor problem is defined by $\mathcal{H}$ and the input is a graph $G=(V,E)$ and a mapping $f : V \to \mathcal{H}; v \mapsto H_v$.)
A spanning subgraph is also known as a factor, hence the name. This framework of problems captures many classic problems. For example, when $H_v = \{1\}$ for all $v \in V$, the problem is to determine if $G$ contains a perfect matching, also called a 1-factor.
Question
What is currently known about the complexity of this problem for different $\mathcal{H}$?
What about the special case when $\mathcal{H}$ is a singleton set (so $H_v$ is the same for all $v \in V$)?
Strongest Results I Know
Tractability: If each set in $\mathcal{H}$ does not contain two consecutive gaps, then the $\mathcal{H}$-factor problem is in P (Cornuejols 1988). A integer $h$ is a gap in $H \subseteq \mathbb{N}$ if $h \not\in H$ but $H$ contains an element less than $h$ and an element greater than $h$.
Hardness: There exist some $\mathcal{H}$ such that the $\mathcal{H}$-factor problem is NP-hard (Lovasz 1972).
See (Szabo 2004) for a modern reference that cites these two results.