We will first review how the subconstituent algebra of a $Q$-polynomial graph is related to a tridiagonal algebra. We then examine a particular tridiagonal algebra, called the positive part $U^+_q$ of the quantum group $U_q({\widehat {\mathfrak{sl}}_2)}$. For $U^+_q$ we describe the Damiani PBW basis, the Beck PBW basis, and the alternating PBW basis. We discuss how these PBW bases are related to each other. We also give these PBW bases in closed form, using a $q$-shuffle algebra. We finish with an open problem.