Does nature of singularity in black hole depend on material that fell in? Electromagnetic waves have a tracesless stress energy tensor, and therefore if they are the only fields in a region of spacetime, the Ricci curvature scalar $R=0$ according to GR.  However $R^{\mu\nu} \ne 0$, and other scalars of curvature can be non-zero, for example ($R^{\mu\nu\rho\sigma} R_{\mu\nu\rho\sigma} \ne 0$).
A massive scalar field provides a stress energy tensor with a non-zero trace.  Therefore $R\ne 0$.
In vacuum solutions the stress energy tensor is identically zero, so for gravity waves $R=0$ and $R^{\mu\nu}=0$.  However $R^{\mu\nu\rho\sigma} \ne 0$, and in particlar the curvature scalar formed from the Weyl curvature can be non-zero ($C^{\mu\nu\rho\sigma}C_{\mu\nu\rho\sigma}$).
All of these (even gravitational waves) can collapse to a black hole.
According to the no-hair theorem externally a blackhole will be characterized only by mass, charge, and angular momentum.  But what about internally?  Considering a collapse with no charge or total angular momentum, it seems like the above three scenarios would give qualitatively different singularities.
 A: I think you're misinterpreting what the no-hair theorems say. There's a no-hair theorem for stationary electrovac solutions. It applies to electrovac solutions, not to solutions containing any matter field you like; in fact, there are known counterexamples if you include certain types of matter fields.
Also, it's a theorem about stationary solutions. A black hole spacetime isn't stationary inside the event horizon. It probably wouldn't make sense to talk about a uniqueness theorem extending inside the horizon, since we'd have to talk about the state of the spacetime at a certain point in time, "now," after it had, e.g., formed by gravitational collapse. But there is no usable notion of "now" that can be extended past the event horizon. One can describe all the infalling matter as only asymptotically approaching the horizon, so it has not yet passed inside the horizon "now."
A complete review article is available here: http://relativity.livingreviews.org/Articles/lrr-1998-6/fulltext.html
A: Black-Hole formation is really an extension of Madame Noether's Conservation Law...where a disorganized mass of particles, in a state of 3 dimensional chaos, becomes subject to a quantum-mechanical process which 'homomorphically' transforms this chaotic state into a 2 dimensional state that has perfect Noether symmetry and equilibrium.  This transformed state consists only of radiation confined to the surface of a Schwarzschild sphere...in other words: a Black-hole. There is no internal energy, including matter, associated with the interior of a Schwarzschild sphere; and there is no singularity.
