Reals which must, can't or might be added by forcing

The paper When is a given real generic over L? by Fabiana Castiblanco and Ralf Schindler characterizes all reals which are generic over $L$.

The characterization mentioned by Mohammad in his answer really dates back to Lev Bukovský in the early 70s, and, as Ralf and Fabiana recognize in their note, has nothing to do with $L$ or with reals (in their note, they indicate that after proving their result, they realized they had essentially rediscovered Bukovský's theorem). See

MR0332477 (48 #10804). Characterization of generic extensions of models of set theory, Fundamenta Mathematica 83 (1973), pp. 35–46.

What Bukovský does is to find a characterization for when $V$ is a $\lambda$-cc generic extension of the given inner model $W$ (for a given regular cardinal $\lambda$). Ralf actually has a nice write-up of Bukovský's theorem (organized differently from the presentation in his note with Fabiana) in his recent paper

The long extender algebra, preprint. To appear in a special issue of Archive for Mathematical Logic.

In fact, it is enough that $W$ uniformly $\lambda$-covers $V$, meaning that for all functions $f\in V$ whose domain is in $W$ and whose range is contained in $W$ there is some function $g\in W$ with the same domain and such that $f(x) \in g(x)$ and $|g(x)| < \lambda$ for all $x \in \operatorname{dom}(f)$.

The generality of the result may be a bit of a drawback for the question at hand, however, since it does not seem to provide us with a widely applicable combinatorial characterization allowing us to identify those reals that are generic over $W$.

The same comment applies to the version of the result described in the note by Ralf and Fabiana. Essentially, given $r$, they look at $W=L$ and $V=L[r]$, and describe whether $W$ uniformly $\lambda$-covers $V$ in terms of possible liftings of elementary embeddings $j\!:L_\alpha\to L_\beta$. The connection with Bukovský's characterization is made explicit in page 7 of their note.