**ADDED (29 May, 2013)**

As has been pointed out in the comments, there has been great progress since this answer was first written, and the conjectures below have now been proved, thanks to ground-breaking work of Agol, Kahn--Markovic and Wise. Here's a brief summary of some of the highlights. (Shameless self-promotion: see this survey article for too many further details, including definitions of some of the terms.)

Haglund--Wise define the notion of special (non-positively curved) cube complex. If a closed hyperbolic 3-manifold $M$ is homotopy equivalent to a special cube complex then $M$ satisfies L (largeness, defined below).

Agol proves that if $M$ is homotopy equivalent to a special cube complex then $M$ also satisfies VFC (the Virtually Fibred Conjecture, also defined below).

Kahn--Markovic prove SSC (the Surface Subgroup Conjecture, also defined below), using mixing properties of the geodesic flow. In fact, they construct enough surfaces to show that $M$ is homotopy equivalent to a cube complex.

Wise proves (independently of Kahn--Markovic) that if $M$ contains an *embedded, geometrically finite* surface then $M$ is special.

Agol uses a very deep theorem of Wise (the Malnormal Special Quotient Theorem) to prove a conjecture (also of Wise), which states that word-hyperbolic fundamental groups of non-positively curved cube complexes are special. All the properties below follow.

It's quite a story, and many other names have gone unmentioned. There were also very important contributions by Sageev (who's thesis initiated the programme of using cube complexes to attack these problems), Groves--Manning, Bergeron--Wise, Hsu--Wise and another very deep paper of Haglund--Wise. To extend these results to the cusped hyperbolic case you need results of Hruska--Wise and Sageev--Wise. Finally, it turns out that similar results hold for all *non-positively curved* 3-manifolds, a result established by Liu and Przytycki--Wise.

Let $M$ be a finite-volume hyperbolic 3-manifold. (Some of these extend, suitably restated, to larger classes of 3-manifolds. But it follows from Geometrisation that the hyperbolic case is often the most interesting. These are all ~~trivial or~~ trivially false in the elliptic case, for example.)

**The Surface Subgroup Conjecture (SSC).** $\pi_1M$ contains a subgroup isomorphic to the fundamental group of a closed *hyperbolic* surface. (Recently proved by Kahn and Markovic.)

**The Virtually Haken Conjecture (VHC).** $M$ has a finite-sheeted covering space with an embedded incompressible subsurface.

**Virtually positive first Betti number (VPFB).** $M$ has a finite-sheeted covering space $\widehat{M}$ with $b_1(\widehat{M})\geq 1$.

**Virtually infinite first Betti number (VIFB).** $M$ has finite-sheeted covering spaces $\widehat{M}_k$ with $b_1(\widehat{M}_k)$ arbitrarily large.

**Largeness (L).** $\pi_1(M)$ has a finite-index subgroup that surjects a non-abelian free group.

**The Virtually Fibred Conjecture (VFC).** $M$ has a finite sheeted cover that is homeomorphic to the mapping torus of a (necessarily pseudo-Anosov) surface automorphism. This is false for graph manifolds. There are fairly easy implications

$L\Rightarrow VIFB \Rightarrow VPFB \Rightarrow VHC \Rightarrow SSC$.

Also, *a fortiori*,

$VFC\Rightarrow VPFB$.

Recently, Daniel Wise announced a proof that $VHC\Rightarrow VFC$. His proof also shows that, if $M$ has an embedded *geometrically finite* subsurface, then we get $L$ and other nice properties.

This list is similar to the one that Agol links to in the comments. Also, I suppose it's exactly what Daniel Moskovich meant by 'The Virtually Fibred Conjecture, and related problems'. I thought some people might be interested in a little more detail.

Paul Siegel asks in comments: 'Would it be correct to guess that the "virtually _ conjecture" problems can be translated into a question about the large scale geometry of the fundamental group?'

Certainly, it's true that most of these can be translated into an assertion about how (some finite-index subgroup of) $\pi_1M$ splits as an amalgamated product, HNN extension or, more generally, as a graph of groups. The equivalence uses the Seifert--van Kampen Theorem in one direction, and something like Proposition 2.3.1 of Culler--Shalen in the other. Rephrased like this, some of the above conjectures turn out as follows.

**The Virtually Haken Conjecture (VHC).** $M$ has a finite-sheeted covering space $\widehat{M}$ such that $\pi_1(\widehat{M})$ splits.

**Virtually positive first Betti number (VPFB).** $M$ has a finite-sheeted covering space $\widehat{M}$ such that $\pi_1(\widehat{M})$ splits as an HNN extension.

**Largeness (L).** $M$ has a finite-sheeted covering space $\widehat{M}$ such that $\pi_1(\widehat{M})$ splits as a graph of groups with underlying graph of negative Euler characteristic.

**The Virtually Fibred Conjecture (VFC).** $M$ has a finite-sheeted covering space $\widehat{M}$ such that $\pi_1(\widehat{M})$ splits can be written as a semi-direct product

$\pi_1(\widehat{M}) \cong K\rtimes\mathbb{Z}$

with $K$ finitely generated. (Here we invoke Stallings' theorem that a 3-manifold whose fundamental group has finitely generated commutator subgroup is fibred.)

I don't think I know a way to rephrase $VIFB$ in terms of splittings of $\pi_1$.

Often, when people say 'the large scale geometry of $\pi_1$' they're talking about properties that are invariant under quasi-isometry. I'm really not sure whether these splitting properties (or, more exactly, 'virtually having these splitting properties') are invariant under quasi-isometry. Perhaps something like the work of Mosher--Sageev--Whyte does the trick?