I agree with grandfather here. Do not cooperate with law enforcement when you have an option. This does not mean be rude to them or try to be difficult. It just means that you have to assert your rights. Any cop who does not respect a person's rights does not, in my eyes, deserve cooperation.
Of course, almost everyone would love to help the police with their investigation as much as they can but in a litigious climate with plea bargaining (and not to even talk about the illegal quotas that officers need to meet), I am simply not convinced that there is enough incentive for law enforcement to get the right person.
Life is not CSI. Cops are people too. They want to go home at the end of the day. I am not saying all police officers are corrupt all the time. I am simply saying most of them are overworked (and perhaps underpaid) a lot. If you think about it from this perspective, I hope you'll agree that not asserting your rights and helping law enforcement is a terrible idea. There is a video on YouTube that talks about the same thing. If they tell you the investigation is not directed towards you, have them give you something in writing (make sure your lawyer reads it) that grants you a full and unconditional pardon. Of course, they will not grant it.
In case of jaywalking, I'd say the law is flawed. If the people who allocate resources for law enforcement think that it is enough of a problem, they should allocate enough resources to cover for people's right to refuse to identify themselves unless presented with a warrant.
In the same vein, I really wish that the monies from fines and tickets not go to the police or to the general treasury at all. I wish we treated such money as bad money. I don't know what we'd do with it but doing any good with bad money taints the good efforts and will eventually rot the effort. (E.g., civil forfeiture in the incredibly stupid war on drugs)
IANAL.