We discuss root-finding algorithms for multiple zeros of nonlinear equations in one variable. Re- cent investigations regarding this problem were mainly aimed at deriving schemes that use the beforehand knowledge of root multiplicity. In this communication we investigate several such root-finding methods under the assumption that the multiplicity of the sought root is not early known. We analyze strategies where root refinement is calculated along side to its multiplicity assessment, and put them to use through numerical experiments. Presented results go in favor of a more realistic use of the analysed methods.