The ability to accurately identify and quantify genetic signatures associated with soft selective sweeps based on patterns of nucleotide variation has remained controversial. We here provide counter viewpoints to recent publications in PLOS Genetics that have argued not only for the statistical identifiability of soft selective sweeps, but also for their pervasive evolutionary role in both Drosophila and HIV populations. We present evidence that these claims owe to a lack of consideration of competing evolutionary models, unjustified interpretations of empirical outliers, as well as to new definitions of the processes themselves. Our results highlight the dangers of fitting evolutionary models based on hypothesized and episodic processes without properly first considering common processes and, more generally, of the tendency in certain research areas to view pervasive positive selection as a foregone conclusion.