Is Floer homology always isomorphic to the singular homology of some space? After I studied Morse homology, I'm now studying the following Floer homology theories : 

1) Symplectic Floer homology ;
2) Floer homology of lagrangians ;
3) Heegard-Floer homology ; 
4) Instanton Floer homology ;
5) Seiberg-Witten-Floer homology ;

I saw that Morse homology of a compact manifold $M$ is isomorphic to singular homology of $M$. 
Similarly, I wonder if 1), 3), 4) and 5) are always isomorphic to singular homology of some corresponding space. If not, under which conditions are they ?
Moreover, does 2) link in some way or another to the others ?
 A: In the case of the Floer homology of the cotangent bundle the answer is yes. You should have a look at this:
The Viterbo transfer as a map of spectra by Thomas Kragh.
A: These are two very different and very nice questions. The correct answer is "probably".
First I should respond to what might be seen as the naive question - "is Floer homology the homology of the manifold we're plugging in? the space of connections? the moduli space of trajectories?" - the answers to all of these are no. The way you would get a space is by patching together these spaces of trajectories somehow, or some sort of Conley index. 
Floer homology - let's say Seiberg-Witten Floer homology - is a functor $\text{Cob}_{3+1} \to \text{GrAb}$. There has been some interest as to whether you can lift this, as you ask, as a functor to spaces. Well, the cobordism maps often induce grading shifts, which wouldn't make sense at the level of spaces; so one asks if you can extend this as a functor to spectra.
As a philosophical point, the way the Morse theory works is that you can attach the unstable manifolds as cells of the appropriate index. This is also why in Morse's original application to loop spaces of Lie groups, he could mostly reconstruct the topology of the loop space: critical points had only finitely many negative eigenvalues of the Hessian, so you're attaching cells for each unstable manifold. In Floer theory, there are always infinitely many positive and negative eigenvalues at a critical point, so that doesn't really make sense. You do have a relative index, and often an absolute grading, so you can say that some critical point should represent a 0-cell, things that differ from it in index by 1 a 1-cell, but also -1-cells etc; this makes sense in the sense of spectra/stable homotopy types. 
The first place you can see people thinking about this idea is in Cohen-Jones-Segal, "Infinite dimensional Morse theory and homotopy theory". (These ideas are perhaps a little bit too rigid - they assume certain smoothness results of moduli spaces which are not always very realistic.) The idea is to take the moduli spaces of trajectories and path them together as stable cells of some spectrum, in much the same way as one patches together spaces of trajectories in Morse theory to get a CW structure on the whole manifold. The resulting spectrum's homology should coincide with the Floer homology.
This program has actually been carried out to a degree in Seiberg-Witten theory (which I should say is isomorphic to Heegaard Floer homology, though this isomorphism is not yet known to be natural; it almost certainly is.) Manolescu constructed a Seiberg-Witten spectrum when $b_1(M) = 0$ here; this year, he showed with Tye Lidman here that the homology of this spectrum really is isomorphic to Seiberg-Witten Floer homology. (I forget if it's known to be naturally isomorphic, but it's again almost certainly true that it is.) He extended it with Kronheimer to the $b_1 \leq 1$ case (when the spin-c structure is nontorsion) and Khandhawit-Lin-Sasahira extended this to all closed oriented 3-manifolds (though there are no cobordism maps yet, and this should be computing Seiberg-Witten-Floer homology with really twisted coefficients). 
To understand these issues, I recommend section 6 in Manolescu's survey here. 
There are serious issues in carrying out Manolescu's (or Cohen-Jones-Segal's) ideas in instanton homology, coming from the bubbling. Abouzaid and Kragh have some work on Floer homotopy types in the symplectic (and Lagrangian) theories, and I think Abouzaid and Manolescu are doing further work on this now, though I'm not particularly familiar with it.

Heegaard Floer homology is defined similarly to the Lagrangian intersection theory (using totally real submanifolds of $\text{Sym}^g(\Sigma_g)$), though Perutz has now shown that it can be defined entirely in terms of Lagrangian intersection homology. The Atiyah-Floer conjecture says that the instanton Floer homology of a homology sphere $\Sigma$ should be the same as the Lagrangian intersection homology of two submanifolds in the representation variety $\text{Hom}(\pi_1 \Sigma_g, SU(2))/SU(2)$ corresponding to a Heegaard splitting of genus $g$ - be warned that this is a singular symplectic manifold, and thus there is no actually well-defined notion of Lagrangian intersection homology. I don't know precisely what you mean by symplectic Floer homology but I would put good odds on it being a special case of Lagrangian Floer homology, too.
