How would you trace the temporary ownership of any particular piece of cash - DNA?

There are sensors sensitive enough to detect drugs, explosives, and perhaps in the near future perhaps DNA can be detected on-site in a very short amount of time. If so, then yes, you could deduce the likely persons who had contact with the drugs. But it would take a central database to correlate the collected information.
How it would work: Randomly, a bank takes in money and runs it through the sensing system. All DNA on the bill is identified and cataloged. Any traces of drugs are identified and cataloged. Any other sensory data of interest is collected and cataloged like pollen, oils, & other chemicals. The money serial number is cataloged (looking for counterfeit money at the same time).

The database is ported to the National Intelligence Service (NIS) and is kept in one of their high security vaults. The system then indexes and sorts all of the data collected. The DNA information is collected and when statistically there are drugs found in high occurrence, then that DNA is associated with an actual person and compared with their occupation.

Investigators are notified of persons with statistically high drugs for their occupation. A Real Estate Management Company will always have drugs on some of its money because many tenants pay in cash. However, the tenant that handled that same piece of cash may always have statistically high occurrence of drugs on their money and they may be a retired person that gets a check each month.

The reason the database would need to be maintained by NIS is that the DNA information could also be used to identify genetic risks associated with the individual. If someone is prone to have diabetes, insurance companies may not ever insure them, or at extremely elevated rates. There are many other similar things that can be ascertained via DNA.

Perhaps as a service to the individual, NIS could provide them with a statistical breakdown of their DNA to be added to the individuals patient records. That way doctors might know that to check for certain allergies based upon statistical probability.

What kind of drugs are you people on anyway?!?! Who cares who had the money? I get mine from the bank... and when I take in the loose bills I'm quite certain I don't want the bank checking the change that I got from a grocery store and assigning my name to the coke they found on it.

Furthermore, I really don't want some corruptable government official fishing for my DNA on everything I touch. Just because you're not guilty of something doesn't mean they can't drag you through the courts. It's liberty and justice for all...if you can afford it. And just because you know you're paranoid doesn't mean they're not out to get you.

Welcome to 1984. While this may be technically feasible, it would be a terrible idea and would destroy any freedom that we have (in the USA, we have the fourth amendment to the constitution for a reason). Mind your own business.

How could you possibly know which possessor of the bill got the coke on it? The bill may have changed hands dozens of times since it was contaminated. Are you going to try to investigate where each contaminated bill has been? The bill may not have ever been in the hands of a drug dealer. Ordinary users are more likely to contaminate the bills. Many users will roll up a bill to use as a straw to snort their coke. A contaminate bill may not have ever even been in the hands of a drug user. It may have picked up the contamination from another bill while sitting in a innocent person's pocket or a cash drawer at the supermarket. Chasing contaminated bills would be futile.

Are you going to try to investigate where each contaminated bill has been?

No, if an individual bill coming from a person has some drugs on it then no big deal, but it is marked against their name in a private file. Something is only done about it when a statistically improbable amount of bills coming from a person have drugs.

How would you know who the bills came from? I don't think drug dealers are going to volunteer to get their money scanned. You would have to collect them from banks or stores or something. But then there is no way of knowing who they came from.

If you want to get the names of possible drug dealers, you can do it very easily by going to drug infested neighborhoods and asking around. Police often know who the dealers are, but they have to catch them with the goods to convict them.