What is the definition of a timelike and spacelike singularity?

A singularity is a condition in which geodesics are incomplete. For example, if you drop yourself into a black hole, your world-line terminates at the singularity. It's not just that you're destroyed. You (and the subatomic particles you're made of) have no future world-lines. A careful definition of geodesic incompleteness is a little tricky, because we want to talk about geodesics that can't be extended past a certain length, but length is measured by the metric, and the metric goes crazy at a singularity so that length becomes undefined. The way to get around this is to use an affine parameter, which can be defined without a metric. Geodesic incompleteness means that there exists a geodesic that can't be extended past a certain affine parameter. (This also covers lightlike geodesics, which have zero metric length.)

There are two types of singularities, curvature singularities and conical singularities.

A black hole singularity is an example of a curvature singularity; as you approach the singularity, the curvature of spacetime diverges to infinity, as measured by a curvature invariant such as the Ricci scalar. Another example of a curvature singularity is the Big Bang singularity.

A conical singularity is like the one at the tip of a cone. Geodesics are incomplete there basically because there's no way to say which way the geodesic should go once it hits the tip. In 2+1-dimensional GR, curvature vanishes identically, and the only kind of gravity that exists is conical singularities. I don't think conical singularities are expected to be important in our universe, e.g., I don't think they can form by gravitational collapse.

Actual singularities involving geodesic incompleteness are to be distinguished from coordinate singularities, which are not really singularities at all. In the Schwarzschild spacetime, as described in Schwarzschild's original coordinates, some components of the metric blow up at the event horizon, but this is not an actual singularity. This coordinate system can be replaced with a different one in which the metric is well behaved.

The reason curvature scalars are useful as tests for an actual curvature singularity is that since they're scalars, they can't diverge in one coordinate system but stay finite in another. However, they are not definitive tests, for several reasons: (1) a curvature scalar can diverge at a point that is at an infinite affine distance, so it doesn't cause geodesic incompleteness; (2) curvature scalars won't detect conical singularities; (3) there are infinitely many curvature scalars that can be constructed, and some could blow up while others don't. A good treatment of singularities is given in the online book by Winitzki, section 4.1.1.

The definition of a singularity is covered in WP and in all standard GR textbooks. I assume the real issue you were struggling with was the definition of timelike versus spacelike.

In GR, a singularity is not a point in a spacetime; it's like a hole in the topology of the manifold. For example, the Big Bang didn't occur at a point. Because a singularity isn't a point or a point-set, you can't define its timelike or spacelike character in quite the way you would with, say, a curve. A timelike singularity is one that is in the future light cone of some point A but in the past light cone of some other point B, such that a timelike world-line can connect A to B. Black hole and big bang singularities are not timelike, they're spacelike, and that's how they're shown on a Penrose diagram. (Note that in the Schwarzschild metric, the Schwarzschild r and t coordinates swap their timelike and spacelike characters inside the event horizon.)

There is some variety in the definitions, but a timelike singularity is essentially what people mean by a naked singularity. It's a singularity that you can have sitting on your desk, where you can look at it and poke it with a stick. For more detail, see Penrose 1973. In addition to the local definition I gave, there is also a global notion, Rudnicki, 2006, which is essentially that it isn't hidden behind an event horizon (hence the term "naked"). What's being formalized is the notion of a singularity that can form by gravitational collapse from nonsingular initial conditions (unlike a Big Bang singularity), and from which signals can escape to infinity (unlike a black hole singularity).

Penrose, Gravitational radiation and gravitational collapse; Proceedings of the Symposium, Warsaw, 1973. Dordrecht, D. Reidel Publishing Co. pp. 82-91, free online at http://adsabs.harvard.edu/full/1974IAUS...64...82P

Rudnicki, Generalized strong curvature singularities and weak cosmic censorship in cosmological space-times, http://arxiv.org/abs/gr-qc/0606007

Winitzki, Topics in general relativity, https://sites.google.com/site/winitzki/index/topics-in-general-relativity

Timelike and spacelike singularities are sets of points in the spacetime where some curvature invariant such as a scalar polynomial constructed out of the Riemann tensor diverges (but all the invariants are finite at all points in the vicinity of the singularity that don't belong to the singularity) so that the nearby points in the set are timelike-separated or spacelike-separated from one another, respectively.

So one may understand what a timelike or spacelike singularity is by understanding the words "timelike", 'spacelike", and "singularity" separately. There's nothing really new in the phrases; the whole is pretty much the sum of its parts. A singularity is a manifold – submanifold of the spacetime – and the spacelikeness and timelikeness is determined just like for any curves or surfaces etc. in the spacetime, from the sign of $ds^2$.

When the dimension of singular set is greater than one, the actual timelikeness or spacelikeness is more complicated and one must talk about the whole signature – number of positive, negative, and null directions in the space. It's still true that when at least some directions along the set are timelike, people will probably call it a timelike singularity although it's a mixed one.