There is a continuous research effort worldwide to improve the reliability, efficiency, and accuracy of numerical computations in various domains. One of the most promising research avenues is to exploit the structural properties of the mathematical problems to be solved. This paper investigates some numerical algorithms for the solution of common and structured eigenproblems, which have many applications in automatic control (e.g., linear-quadratic optimization and H-infinity-optimization), but also in various areas of applied mathematics, physics, and computational chemistry. Of much interest is to find the eigenvalues and certain deflating sub-spaces, mainly those associated to the stable eigenvalues. Several simple examples are used to highlight the pitfalls which may appear in such numerical computations, using state-of-the-art solvers. Balancing the matrices and the use of condition numbers for eigenvalues are shown to be essential options in investigating the behavior of the solvers and problem sensitivity.