I know that not all disputes are created equal, but what should I be asking or looking for to determine the seriousness of the situation? What situations/disputes are more serious than others?
There are too many disputes to mention or speculate what your response should be. The two questions that would come to my mind are,  is he open and honest so I feel assured I can trust him, and  are my investments safe [a.k.a. Bernie Madoff]. We wrote a white paper on how to know your assets are safe. The key points were that your investments should be with a custodian independent of your advisor, that you should be able to go to the custodian’s website and see your investments which should be publicly traded investments that can be found on one of the financial websites such as Yahoo’s. your investment account statements should also come from that well-known independent custodian, not the advisor. If you have that scenario, even if the guy loses his license or in an extreme case should go to jail, your investments should be safe. If you would like to get in contact, I would be happy to give a more personalized opinion.
Though I haven't had client or regulatory problems during my 25+ year career, I don't think a dispute-free record is a pre-requisite to selecting an advisor. I know many fine advisors with a blemish or two on their record.
Your best course of action is to research the FINRA and SEC websites for advisor information, and if a customer or regulatory issue comes up, don't be shy to ask the advisor to explain. Most are only too happy to give you their side of the story. Advisors usually have good sales skills, so you can expect their explanation to be often satisfying, so look for patterns in the record and the explanations. If you don't like what you hear, or you see a pattern of unhappy clients or regulatory mishaps, keep looking for a different advisor.
Keep in mind, however, that bear markets often bring lawsuits, as clients don't like losing money! Often brokerage firms will settle suits just to move forward, even if the advisor would rather fight the claim. FYI, simple poor investment performance is not fair grounds for a lawsuit.
This is a great question. All disputes are reported and if you look at the FINRA and SEC sites there will be a description of each event. I would look to see if there is a pattern or if it is a onetime event. See how long ago the dispute occurred and this is key see what the advisor had to pay out of pocket. If the advisor did not pay anything then there was probably little cause. If on the other hand the advisor had to pay out of pocket a settlement then there is probably more to the story and I would suggest asking the advisor up front to explain the dispute and if they cannot communicate clearly what happened and what the end results were then I would avoid them. In this world of looking for a scapegoat for all mistakes good advisors may have one or two disputes. In full disclosure I have one dispute in my 19 year career from over 12 years ago. The client was made whole over an input error. The client was valid in complaining and I worked with my employer to make sure that the client received what was promised. We live in a world of complicated products and services. Get everything in writing and make sure you completely understand what you are purchasing. The other thing to remember is that just because there is not a complaint does not mean the advisor is honest. This is probably the main point of your question. Madoff did not have a bad record. Finally I always encourage clients to trust but verify. If you have any question about what your advisor suggests ask someone else for their opinion.
I agree with the responses that you should feel absolutely comfortable in asking a potential investment advisor what is the truth behind the regulatory dispute. If you cannot find comfort in the response you are provided, move on to another advisor. You will find that some very serious issues will be clear while other issues might have a reasonable explanation. But, if there are several issues or one that clearly is a serious issue, you would be wise to have your guard up.