In general relativity, a naked singularity is a hypothetical gravitational singularity without an event horizon. In a black hole, the singularity is completely enclosed by a boundary known as the event horizon, inside which the gravitational force of the singularity is so strong that light cannot escape. Hence, objects inside the event horizon—including the singularity itself—cannot be directly observed. A naked singularity, by contrast, would be observable from the outside.
Source : Wikipedia
The existence (or not) of naked singularities has not been proved or disproved - either by experimental investigations, cosmological observations, or by theoretical calculations.
The 'weak cosmic censorship hypotheses' and the 'strong cosmic censorship hypotheses' are two mathematical conjectures (from the 1960s) by Roger Penrose which suggest that such singularities can't exist. Other researchers (notably the late Stephen Hawking) maintain that they can.
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