Lesson 1: Ethical standards rather than complex rules.
If you do not believe that the people who work with and for you have high ethical standards, there is no point in setting up complex rules to try to control them. Standards and principles work,in my opinion, although it may not be obvious why. There is often a delight in writing new rules to cover new situations. The difficulty with specific rules is that, if there are many of them, the lesson for staff can be "if there is no rule against it, then you can do it". By having a simple principle such as "Behave responsibly" or "Act in the best interest of the client" there is enough vagueness to persuade staff to err on the side of caution.
I worked with one organisation which moved from a set of reasonably well understood financial principles to a 60 page set of financial regulations. Financial efficiency went downhill - partly through the complexity of the rules and partly because of the "work to the rules and only the rules" way of working.
Lesson 2: If you have done wrong - admit it and go immediately
We can all make mistakes. If a member of staff - no matter how senior - has "made a mistake with their expenses" for example, they should admit the error and then resign. This may sound as if the punishment is more serious than the "crime", however the alternative is to say "it's ok to fake some of your expenses so long as you are an important employee and we don't think it's too serious". To me this sounds rather like tempting staff to take a chance. Zero tolerance has the advantage of being easy to understand.
I worked in a hotel company which had a published zero tolerance for drinking on duty. Every member of staff knew about the rule but we didn't think it was that serious. One Saturday evening the General Manager was acting as Duty Manager, and late one night was standing at the bar drinking a small glass of beer. The Area Manager happened to visit, saw the GM drinking and fired him on the spot. We had no problem convincing staff that the rule was serious after that.
Lesson 3: Transparency is easier
The more things are kept private and secret, the more people are convinced that the reason for the secrecy is because they are being disadvantaged.
Staff expenses should be available throughout a firm (and to the shareholders or partners depending on the structure) - this leads to better behaviour from everyone since they will not only need to justify their expenses to their line manager, but to the firm as a whole. Why should anyone be ashamed of legitimate expenses? The same can be applied to most expenditure. Even salaries can be published by way of salary bands - it works in the UK Civil Service, so why not everywhere.
By considering lessons for our own workplaces, perhaps some good can come out of the mess that our MPs have made of theirs.