To provide a fibrous single crystal having smooth surface form, excellent uniformity and a diameter of <100 μm and provide its production process.

The objective fibrous single crystal is composed of either one of LiNbO3, LiTaO3, TiO2, α-Al2O3, Y3Al5O12 and Si and has a crystal diameter of <100 μm. The process for the production of the fibrous single crystal comprises a step to heat a crucible 5 filled with the single crystal raw material and provided with a capillary pipe 9 having a needle-like material and protruded on the bottom and melt the single crystal raw material, a step to contact the molten liquid in the capillary pipe 9 with the seed crystal and a step to draw the molten liquid attached to the seed crystal downward to effect the cooling and the solidification of the liquid.