The boundary element method (BEM) is used to analyse the singular behaviour at the free edges of fibre-matrix interface in the fibre push-out test. The fibre push-out test has been widely employed to characterise the interface properties in polymer, ceramic and metal matrix composites. There are two free edges in the fibre push-out test: one is at the loaded fibre end and the other at the supported fibre end, where both the shear and radial interface stress components exhibit singularity. Numerical study of the effect of different mesh sizes confirms that the magnitudes of stresses obtained using sufficiently fine meshes are real and accurate along the whole interface, except the regions very close to the singular points. The BEM is applied to calculate these singular stresses, which are a function of the singular exponent, α, and the singular stress intensity, Fij. It is shown that the singular exponents obtained at both the loaded and supported fibre ends are characteristic of composite constituents properties, such as Young's moduli of fibre and matrix. Of note is that the singularity exponents are the same for the interface shear and radial stress components at fibre ends, because the wedge angles are the same. They are real in the fibre push-out model because the interface stress profiles do not contain oscillation functions. The singular stress intensities, Fij, are finite, implicit of material properties, and vary with specimen geometry and dimensions, such as the fibre to matrix radius ratio, fibre aspect ratio and support hole size, with the general trends depending on the location of singular point. It is also noted that the absolute values of the singular stress intensity ratio, Fr/Frz, obtained for the loaded and supported fibre ends display functional analogy with respect to certain material and geometric parameters, such as fibre to matrix Young's modulus ratio, Ef/Em, and fibre to matrix radius ratio, a/b.