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

<cite authors="Voigt, Thomas; Ziegler, Günter M.">_Voigt, Thomas; Ziegler, Günter M._, [**Singular 0/1-matrices, and the hyperplanes spanned by random 0/1-vectors**](http://dx.doi.org/10.1017/S0963548305007339), Comb. Probab. Comput. 15, No. 3, 463-471 (2006). [ZBL1165.15018](https://zbmath.org/?q=an:1165.15018).</cite>
  [1]: https://arxiv.org/abs/math/0308050