Directional routing of one-way classical wave has raised tremendous interests about spin-related phenomena in topological metamaterials. This sparks specifically the elastic wave study of synthesizing pseudo-spin degree-of-freedom in meta-structures for implementing topological phononic devices to perform robust elastic wave manipulations. Unlike pseudo-spin in mathematical sense, the physically intrinsic spin angular momentum of elastic wave is predicted quite recently which exhibits selective excitation of unidirectional wave propagation even in conventional solids. However, due to the grand challenge of building up chiral elastic sources, the experimental observation of intrinsic spin of elastic wave and relevant properties is still missing. Here, we successfully measure the elastic spin in two typical elastic modes, i.e. Rayleigh and Lamb waves, by adopting the elaborately designed chiral meta-sources that excite locally rotating displacement polarizations. In both systems, we observe the unidirectional routing of chiral elastic waves, characterize the different elastic spins along different directions, and demonstrate the spin-momentum locking in broad frequency ranges. We also find the selective one-way Lamb wave carries opposite elastic spin on two plate surfaces in additional to the source chirality. The observation of elastic spin and related intriguing phenomena paves a new way for chiral elasticity, miniature on-chip devices, and spin-sensitive sensors.