Spherical nucleic acids have emerged as a class of nanostructures, exhibiting a wide variety of properties, distinctly different from those of linear nucleic acids, and a plethora of applications in therapeutics and diagnostics. Herein, we report on preparation of 3D nucleic acid nanostructures, prepared by self-assembly of polymer-oligonucleotide conjugates. The latter are obtained by grafting multiple alkyne-functionalized oligonucleotide strands onto azide-modified homo-, block, and random (co)polymers of chloromethylstyrene via initiator-free click coupling chemistry to form conjugates of comblike and coil-comb chain architectures. The resulting conjugates are amphiphilic and form stable nanosized supramolecular structures in aqueous solution. The nanoconstructs are thoroughly investigated and a number of physical characteristics, in particular, molar mass, size, aggregation number, zeta potential, material density, number of oligonucleotide strands per particle, grafting density, and their relation to hallmark properties of spherical nucleic acids - biocompatibility, resistance against DNase I, cellular uptake without the need for transfection agents - are determined.