Main results include: 1) A system of invariants of knots, links, and plane curves, stronger than all known polynomial invariants; 2) A universal method of the computation of homology groups of spaces of non-degenerate geometrical objects, producing in particular the knot invariants and numerous comparison theorems of the Smale–Hirsch–Gromov type for spaces of real and complex functions and maps without complicated singularities; 3) Best known (and asymptotically sharp) estimates of the numbers of branchings of algorithms of approximate computation of roots of polynomials 4) "Stratified" version of the Picard–Lefschetz theory for the homology groups of singular manifolds; 5) A proof of the Atiyah–Bott Garding conjecture on the equivalence of the sharpness of the wave front of a hyperbolic operator to the topological Petrovskii condition; an interpretation of these conditions in the terms of the geometry of the front; 6) Multidimensional analogues of the Newton's theorem on the non-integrability of ovals; 7) A construction of multi-dimensional analogues of the Maslov index by the methods of the singularity theory (the "universal complex of singularities"); 8) A calculation of the stable homotopy types of complements of plane arrangements in $\mathbf R^n$, generalizing the Goresky–MacPherson formula for the homology groups of such complements. Other results concern complexity theory of smooth maps, real algebraic geometry, generalized hypergeometric functions, topology of Lie groups, dynamical systems, geometrical combinatorics, potential theory, etc.
Main publications:
V. A. Vassiliev, Complements of discriminants of smooth maps: topology and applications, 2-d extended edition, Translations of Math. Monographs, 98, AMS, Providence, RI, 1994, 268 pp. ; rasshirennyi russkii perevod: V. A. Vasilev, Topologiya dopolnenii k diskriminantam, Fazis, Moskva, 1997, xiv+536 s.
V. A. Vassiliev, Ramified integrals, singularities and lacunas, Math. Appl., 315, Kluwer Academic Publishers, Dorderecht (Netherlands), 1995, xviii+289 pp. ; rasshirennyi russkii perevod: V. A. Vasilev, Vetvyaschiesya integraly, MTsNMO, Moskva, 2000, 432 s.
V. A. Vassiliev, Lagrange and Legendre characteristic classes, 2-d edition, Gordon and Breach Publishers, New York a.o., 1993, 273 pp. ; rasshirennyi russkii perevod: V. A. Vasilev, Lagranzhevy i lezhandrovy kharakteristicheskie klassy, MTsNMO, Moskva, 2000, 312 s.
V. I. Arnold, V. A. Vasilev, V. V. Goryunov, O. V. Lyashko, Osobennosti. I. Lokalnaya i globalnaya teoriya, Itogi nauki i tekhniki. Sovr. probl. matem. Fundam. napr., 6, VINITI, Moskva, 1988, 256 s. ; Osobennosti. II. Klassifikatsiya i prilozheniya, Itogi nauki i tekhniki. Sovr. probl. matem. Fundam. napr., 39, 1989, 256 s. ; English translation: V. I. Arnold, V. A. Vasil'ev, V. V. Goryunov, O. V. Lyashko, Singularities, Encycl. Math. Sci., 6, 39, Springer-Verlag, Berlin–New York, 1993, 245+233 pp.
V. A. Vassiliev, “Combinatorial formulas for cohomology of spaces of knots”, Moscow Math. J., 1:1 (2001), 91–123
; rasshirennyi russkii perevod: V. A. Vasilev, Topologiya dopolnenii k diskriminantam, Fazis, Moskva, 1997, xiv+536 s. 
