Flexible polyelectrolytes of different sizes cannot be separated by free solution electrophoresis beyond a certain length exceeding a few monomers. The reason for this size-independent mobility is the equal scaling of the drag and driving forces with the molecular size. By selectively modifying the polyelectrolytes, this scaling symmetry can be broken and a size-dependent mobility can be achieved. We investigate the mobility of short flexible polyelectrolytes, such as single-stranded DNA, with and without a neutral label the size of a single nucleic acid. Numerical simulations allow us a systematic study of the microscopic dynamics of these systems and the impact of the neutral label, which is used to break the initial scaling symmetry, on the mobility. We find a free rotation of all the polyelectrolytes under investigation due to thermal fluctuations (this is in contrast to double-stranded DNA, where the fluctuations are less important due to its larger mass and charge per persistence length and where alignment with the external field has been observed). The impact of the neutral label can be observed as a systematic decrease of the mobility which becomes less pronounced for increasing total molecular size. This suggests that attaching a short neutral label will not allow for single base resolution of single-stranded DNA in free solution. On the other hand, the shift of the mobility due to the neutral label can be significant enough to utilize this method as diagnostic tool by specifically labeling only single-stranded DNA fragments with desired properties. Free solution electrophoresis experiments using single-stranded DNA fragments and biotin and trimethoxytrityl modifications as neutral labels confirm the numerical results.