Heegaard Floer Homology of double branched cover The question is the following:
Let $K\subset S^{3}$ be a knot, consider the double branched cover $Y$ of $S^{3}$ over $K$. We know $Y$ has a unique spin structure $\mathfrak{s}$, now the question is: 
When do we have $\widehat{HF}(Y,\mathfrak{s})\cong \mathbb{Z}$? If yes, what do we know about the d-invariant?
I know that this is true for quasi-alternating knot, however, I don't know any other examples...
 A: The paper of Manolescu and Owens "A concordance invariant from the floer homology of double branched covers" seems to answer your question.  They compute many examples of the Froyshov/d-invariant in their paper, using software of Saso Strle.  It's available on the arXiv.  In that case it's at least reasonably-computable. 
A: Let assume $Y$ is a closed oriented rational homology sphere.
The set of spin$^c$ structure of $Y$ is in one to one correspondence with $H_1(Y,\mathbb{Z})$. So if $H_1(Y,\mathbb{Z}) \neq 0$ then there are more than one spin$^c$ structure. 
The question then becomes:  When does $\widehat{HF}(Y,\mathfrak{s}) \cong \mathbb{Z}$ for each spin$^c$ structure $\mathfrak{s}$? 
Three-manifolds with such property are called $L$-spaces. Double branched cover of quasi-alternating knots (in $S^3$) are L-spaces. I think your question is a work in progress. For some other examples see   Steven Sivek  comment below.
Note also that the collection of quasi-alternating knots contains the collection of alternating knots, and  2-bridge knots are examples of alternating knots.
In the special case  $H_1(Y,\mathbb{Z}) = 0$, in which situation there is a unique spin$^c$ structure, it is conjectured that $S^3$ and the Poincar\'e homology sphere ($\Sigma=S^3/I$ where $I$ is the full icosahedral group) are the only such 3-manifold. We know that $d(S^3)=0$ and that $d(\Sigma)=\pm 2$ depending on the orientation you chose. For the general case I think knowing exactly the $d$-invariant is still an open problem except when $Y$ is a lens space in which case there is a formula for each $d$ invariant.
A: John Baldwin said something about both questions in this paper. He considers closures of 3-braids and classifies which of these have an $L$-space branched double cover (Theorem 4.1). He then computes the correction term for the spin structure (Theorem 5.1); he also seems to say that there are no new results in this last computation (see the discussion before the statement of the theorem).
