Let $M$ be a smooth $d$-dimensional oriented Riemannian manifold, and let $p \in M$. Let $0 \le k \le d$ be fixed.
Does there exist an open neighbourhood $U$ of $p$, which admit a non-zero harmonic $k$-form? i.e $\omega \in \Omega^k(U)$ satisfying $d\omega=\delta \omega=0$?
Since a form $\omega$ is harmonic if and only if $\star \omega$ is harmoinc ($\star$ is the Hodge dual operator), the answers for a given $k,d-k$ are the same.
For $k=d$, one can take $\omega$ to be the Riemannian volume form.
For $k=1$, we can take $\omega=df$ where $f$ is a harmonic function. Then $\delta \omega=\delta df=0$. Locally, there are always harmonic functions- we can solve the Dirichlet problem on a small ball with boundary, that is finding a harmonic function which is zero on the boundary.
This solves the cases $k=0,1,d-1,d$.
So, we are left with the cases $2 \le k \le d-2$.