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One of the most significant aspects of crystal chemistry of multianionic oxyhydrides is the possibility of flexible regulation of the composition-structure-function relationships. In the context of competitive coordinations of different anions in the crystal lattice, this may afford formation of a number of stable stoichiometric phases without inversion symmetry. In the present work, we demonstrated that semiconducting yttrium and lantanium oxyhydrides with the composition Ln2H4O (Ln=Y, La) have an attractive potential for the design of novel lead-free ferro- and piezoelectric systems. By means of advanced DFT-based computational simulations we predicted that several polar monoclinic and orthorhombic phases of Ln2H4O may exhibit exceptional ferro- and piezoelectric properties as well as electromechanical coupling characteristics that are especially suitable for the piezoelectric devices working in a shear mode. Structure-dependent theoretical evaluations of the relevant physical responses demonstrated estimates of ferro- and piezoelectric characteristics that are comparable with the specifications of advanced ferroelectric solid solutions. Thus, our prediction of lead-free piezoelectric systems forms a solid and technologically reliable basis for the future development of effective and non-hazardous materials.