In this paper, a highly-sensitive distributed shape sensor based on a multicore fiber (MCF) and phase-sensitive optical time-domain reflectometry (φ-OTDR) is proposed and experimentally demonstrated. The implemented system features a high strain sensitivity (down to ∼0.3 µɛ) over a 24 m-long MCF with a spatial resolution of 10 cm. The results demonstrate good repeatability of the relative fiber curvature and bend orientation measurements. Changes in the fiber shape are successfully retrieved, showing detectable displacements of the free moving fiber end as small as 50 µm over a 60 cm-long fiber. In addition, the proposed technique overcomes cross-sensitivity issues between strain and temperature. To the best of our knowledge, the results presented in this work provide the first demonstration of distributed shape sensing based on φ-OTDR using MCFs. This high-sensitivity technique proves to be a promising approach for a wide range of new applications such as dynamic, long distance and three-dimensional distributed shape sensing.