First-principles calculations have become a powerful tool to exclude the Edisonian approach in search of novel two-dimensional (2D) materials. However, no universal first-principles criteria to examine the realizability of hypothetical 2D materials have been established in the literature yet. Because of this, and as the calculations are always performed in an artificial simulation environment, one can unintentionally study compounds that do not exist in experiments. Although investigations of physics and chemistry of unrealizable materials can provide some fundamental knowledge, the discussion of their applications can mislead experimentalists for years and increase the gap between experimental and theoretical research. By analyzing energy convex hull, phonon spectra, and structure evolution during ab initio molecular dynamics simulations for a range of synthesized and recently proposed 2D materials, we construct energy, phonon, and dynamic stability filters that need to be satisfied before proposing novel 2D compounds. We demonstrate the power of the suggested filters for several selected 2D systems, revealing that some of them cannot be ever realized experimentally.