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The Universe's densest objects are black holes. In the second place, there are neutron stars.

So, if a neutron star compresses to its Schwarzschild radius, would it appear as a black hole? That black hole would be one of the most dense objects in the universe?

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Actually, before you got to the Schwarzschild radius, the pressure previously supporting the neutron star against its own gravity would no longer be able to do so, and the entire star would violently collapse. Some matter would be thrown out, while the rest would become a black hole. The singularity at the center would be among all the rest of the singularities at the centers of all other black holes for the densest objects in the Universe.

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Consider a "stable" neutron star. Are there any mechanisms that allow it to slowly compress further, until it's enough to implode into a singularity? Would mere cooling do that? –  Florin Andrei Mar 22 '12 at 16:36
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@FlorinAndrei Neutron stars are supported by degenerate neutron pressure. Converting neutrons into strange hadrons would reduce the neutron density, but if that mechanism is energetically favorable you get a soft equation of state and consequently a low maximum mass which increasingly looks ruled out by the data. –  dmckee Mar 22 '12 at 18:58
    
So, that seems to indicate that, if they don't collapse into a black hole at first, chances are won't collapse later on, is that right? –  Florin Andrei Mar 22 '12 at 19:56
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Unless there is an external source of mass...as is the case in some binary systems that include neutron stars. Mass loss from the companion accumulates on the neutron star until BOOM you get a type uhm...one? supernova. –  dmckee Mar 22 '12 at 20:37
    
Hmmm... There's a thought. Type Ia supernova occur when a carbon-oxygen white-dwarf gains enough mass to start fusing carbon into heavier elements. I'm not sure if a parallel exists for accreting neutron stars or, if not, why not. Perhaps time for a new question... –  Warrick Mar 23 '12 at 8:36
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Just to add a little (especially to @dmckee's comment to @Florin_Andrei's response):

A typical, isolated neutron star stably resists gravity with neutron degeneracy pressure. If instead it is accreting mass from a binary companion, it may grow beyond the Tolman-Oppenheimer-Volkoff Limit (just like the Schwarzschild limit for white-dwarfs, except for neutron stars), at which point it will inevitably succumb to gravity and collapse.

Neutron-star collapse is believed to almost always form a black-hole remnant; the exact details are unknown, but there are numerous quite successful models. What exactly it would look like is unclear, but the emission would be much less than the supernova of a typical massive star, or white dwarf. The observational signature would be entirely unlike a type Ia, and most likely unlike a type II as-well.

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Type II supernovae are from the collapse of the cores of massive stars, and have to work their way out from the heavy blanket of material above the core. So yeah, a neutron star collapse will look entirely different than that. –  Andrew Apr 6 '12 at 11:04
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