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Numerical procedures and simulation techniques in science and engineering have progressed significantly during the last decades. The finite element method plays an important role in this development and has gained popularity in many fields including fluid mechanics. A recent finite element solution strategy is isogeometric analysis. Isogeometric analysis replaces the usual finite element basis functions by higher-order splines. This leads to significantly more accurate results and equips the numerical method with several desirable properties. By naively applying the finite element isogeometric method one may obtain solutions that are seriously perturbed and are as such not physically relevant. The reason is often linked to the stability of the method; a finite element method is not a priori stable. The overall objective of this thesis is centered around this point. The aim is to develop numerical techniques that inherit the stability properties of the underlying physical system. In particular we are interested in finite element techniques that can be applied to free-surface flow simulations. Stability issues in free-surface flow computations may already appear in single-fluid flow problems. Other causes of instabilities are steep layers or discontinuities and instabilities arising from the numerical treatment of the interface that separates the fluids. This thesis addresses each of these topics. Several stabilized finite element methods