This paper presents a non-iterative phase retrieval method from randomly phase-shifted fringe images. By combining the hyperaccurate least squares ellipse fitting method with the subspace method (usually called the principal component analysis), a fast and accurate phase retrieval algorithm is realized. The proposed method is simple, flexible, and accurate. It can be easily coded without iteration, initial guess, or tuning parameter. Its flexibility comes from the fact that totally random phase-shifting steps and any number of fringe images greater than two are acceptable without any specific treatment. Finally, it is accurate because the hyperaccurate least squares method and the modified subspace method enable phase retrieval with a small error as shown by the simulations. A MATLAB code, which is used in the experimental section, is provided within the paper to demonstrate its simplicity and easiness.