Finely divided, starch-containing polymer dispersions which are obtainable by free radical emulsion copolymerization of (a) from 25 to 50% by weight of at least one optionally substituted styrene, methyl methacrylate, acrylonitrile and/or methacrylonitrile, (b) from 1 to 49% by weight of at least one C1-C4-alkyl acrylate and/or one C2-C4-alkyl methacrylate, (c) from 1 to 49% by weight of at least one C5-C22-alkyl acrylate and/or one C5-C22-alkyl methacrylate and (d) from 0 to 10% by weight of at least one other ethylenically unsaturated copolymerizable monomer and (e) from 15 to 40% by weight of a degradable starch which has a molar mass Mw of from 1000 to 65 000 the sum (a)+(b)+(c)+(d)+(e) being 100% and being based on the total solids content, in the presence of redox initiators and at least 0.01% by weight, based on the monomers used, of at least one polymerization regulator, processes for the preparation of the finely divided, starch-containing polymer dispersions by emulsion polymerization of the monomers (a), (b), (c) and (d) with redox initiators in the presence of (e) degraded starch and at least 0.01% by weight, based on the monomers used, of at least one polymerization regulator, and use of the finely divided, starch-containing polymer dispersions as sizes for paper, board and cardboard.