Thermoplastic molding compositions, based on polymers of a monovinyl-aromatic compound which have been modified with rubber to increase their impact strength, which also exhibit increased stress crack resistance.The increased stress crack resistance is achieved if the proportion of the soft component in the polymer which has been modified to increase its impact strength, and the particle size and swelling index of the soft component phase, conform to particular values, and the tensile strength and yield stress of the thermoplastic molding compositions bear a particular ratio to one another.The molding compositions may in particular be used for the manufacture of internal liners of refrigerators and foodstuff packaging containers.