For d=1, this is covered in Karatzas-Shreve in the section on Feller-tests. In particular, in exercise 5.38, they have that if $$\int_{x-\epsilon}^{x+\epsilon}b^{2}(y)dy<\infty,$$

then $dX_{t}=b(X_{t})dt+dW_{t}$ has a weak solution up to explosion time $S$ and it is unique in probability law. The main idea is using Girsanov theorem (hence that square-integrability condition).

For $d\geq 1$, see "Strong solutions of stochastic equations with singular time dependent drift" and "strong solutions of stochastic differential equations with square integrable drift", where again they require integrability: there exists open set $Q\subset\mathbb{R}^{d+1}$

$$\int \left(\int |b(t,x)1_{Q}(t,x)|^{p} dx\right)^{q/p}dt<\infty,$$

for some $p\geq 2, q>2$ with $\frac{d}{2}+\frac{2}{q}<1$.