  We use the theory of Gr\"obner-Shirshov bases for ideals to construct linear
bases for graded local Weyl modules for the (hyper) current and the truncated
current algebras associated to the finite-dimensional complex simple Lie
algebra $\mathfrak{sl}_2$. The main result is a characteristic-free
construction of bases for this important family of modules for the hyper
current $\mathfrak{sl}_2$-algebra. In the positive characteristic setting this
work represents the first construction in the literature. In the characteristic
zero setting, the method provides a different construction of the
Chari-Pressley (also Kus-Littelmann) bases and the Chari-Venkatesh bases for
local Weyl modules for the current $\mathfrak{sl}_2$-algebra. Our construction
allows us to obtain new bases for the local Weyl modules for truncated current
$\mathfrak{sl}_2$-algebras with very particular properties.
