This paper presents a vector expression of the constant-orientation singularity locus of the general Gough–Stewart platform. The third-degree vector expression obtained does not contain a constant term, which allows the factorization of an instance of the position vector, thereby leading to a very compact form. Additionally, an expression of the vector orthogonal to the singularity locus is obtained as a byproduct. An alternative expression that reduces the number of times that the position vector appears in the expression is also presented. It is shown that a simplified architecture such as that of the Minimal Simplified Symmetric Manipulator (MSSM) can significantly reduce the complexity of the coefficients appearing in the expression.