The only practical way to assess DNA and RNA sample purities is via spectrophotometer/Nanodrop readings:
Pure DNA samples show a 260nm/280nm absorbance ratio between 1.8 and 2.0. The 260nm/230nm ratio should be above 2.0
Pure RNA samples show a 260nm/280nm absorbance ratio between 1.8 and 2.1. The 260nm/230nm ratio should be above 1.5
Please note that this method only generates reliable data for DNA and RNA concentrations above 15 to 20 ng/μl. Lower concentration samples almost always show skewed absorbance ratios.
Unfortunately there is no practical way to verify the purity for lower concentration samples. Please see this TechNote from NEB for additional data on this topic.
Thus, we have to rely on best practises during sample isolation to facilitate successful experiments. We are happy to discuss the methods. Please see the consultation links.