In a lab setting, a simple restriction digest and recombination can yield a modification with no signature, or else you can generate SNPs with PCR. And you would want to, as viral genomes are so small, often with overlapping ORFs, that you don't have much room to leave a bunch of junk around.
There is no signature. What we can do is look at sequence similarity and decide how probable or improbable a given sequence combination is from the viral genome sampling available. It's a lot of handwavy Bayesian statistics, based on incomplete sampling.
Instead what they do is look at the animal population in the region for highly related or identical versions. If it was natural transmission you would expect animals to be infected with a strain that is extremely similar to the virulent strain.
What this is saying is that there has been no discovery of an intermediate host which has a highly related genome. This would happen if the version that spread in humans had been incubated in a lab for a long time and had undergone enough mutations in the lab to be very dissimilar to natural versions.
<<Investigators now face the reverse situation for tracing the origins of SARS-CoV-2: a paucity of evidence and a cold trail. Closely related bat SARSr-CoVs have been identified, but there is still no sign of an intermediate host 1.5 years into the pandemic. The closest bat virus relatives have been found in Yunnan province about a thousand miles away from Wuhan in Hubei province [17–19]. >>
