Constructing a highly localized wave field by means of bound states in the continuum (BICs) promotes enhanced wave-matter interaction and offers approaches to high-sensitivity devices. Elastic waves can carry complex polarizations and thus differ from electromagnetic waves and other scalar mechanical waves in the formation of BICs, which is yet to be fully explored and exploited. Here, we report the investigation of local resonance modes supported by a Lamb waveguide side-branched with two pairs of resonant pillars and show the emergence of two groups of elastic BICs with different polarizations or symmetries. Particularly, the two groups of BICs exhibit distinct responses to external perturbations, based on which a label-free sensing scheme with enhanced-sensitivity is proposed. Our study reveals the rich properties of BICs arising from the complex wave dynamics in elastic media and demonstrates their unique functionality for sensing and detection.