Starting with a brief review of our prior construction of $n$-ary algebras in noncommutative Clifford spaces, we proceed to construct in full detail the Clifford-Yang algebra which is an extension of the Yang algebra in noncommutative phase spaces. The Clifford-Yang algebra allows to write down the commutators of the $noncommutative$ polyvector-valued coordinates and momenta and which are compatible with the Jacobi identities, the Weyl-Heisenberg algebra, and paves the way for a formulation of Quantum Mechanics in Noncommutative Clifford spaces. We continue with a detail study of the isotropic $3D$ quantum oscillator in $noncommutative$ spaces and find the energy eigenvalues and eigenfunctions. These findings differ considerably from the ordinary quantum oscillator in commutative spaces. We find that QM in noncommutative spaces leads to very different solutions, eigenvalues, and uncertainty relations than ordinary QM in commutative spaces. The generalization of QM to noncommutative Clifford (phase) spaces is attained via the Clifford-Yang algebra. The operators are now given by the generalized angular momentum operators involving polyvector coordinates and momenta. The eigenfunctions (wave functions) are now more complicated functions of the polyvector coordinates. We conclude with some important remarks.