A nice survey is given by [Voigt and Ziegler][1] (but they only address the $p=\frac12$ case explicitly).

Voigt, Thomas; Ziegler, Günter M., Singular 0/1-matrices, and the hyperplanes spanned by random 0/1-vectors, Comb. Probab. Comput. 15, No. 3, 463-471 (2006). ZBL1165.15018. [1]: https://arxiv.org/abs/math/0308050

A nice survey is given by Voigt and Ziegler (but they only address the $p=\frac12$ case explicitly).

Voigt, Thomas; Ziegler, Günter M., Singular 0/1-matrices, and the hyperplanes spanned by random 0/1-vectors, Comb. Probab. Comput. 15, No. 3, 463-471 (2006). ZBL1165.15018.