From what I know, they all mean similar things (way, method), but 方式 can also mean style which makes it similar to 样式. Is there a difference in the way they are used grammatically or is the difference mainly about preference?
I haven't looked up in the dictionary, so my answer may not be accurate. Just for reference.
方法 means the way that should be followed when doing something. Example: 使用方法 (the way to use something, like washing machine or ointment)
办法 seems to be more colloquial, and mostly refers to the way to solve a problem or achieve a goal. Sometimes it is equivalent to 方法. Example: 我没办法吃这么多。(I can't make it to eat this much. )
方式 refers to the "style" when doing something. However, it is not eqivalent to 样式. The latter refers to what a thing looks like. Example: 传统方式未必是落后的。(Traditional way(style) is not definitely out of date. ) And: 传统的样式是很好看的。(Traditional style is very beautiful. )
Hope this will help you. ^o^
方法 refers to a method in the strict sense of a careful organized plan that controls the way something is done. This is also expressed by tiáolǐ 条理 -- you follow a known rule.
办法 means you somehow get the problem solved, but maybe not by any rule. Maybe you make up a new, temporary solution, that no one else has ever tried, and you will never do again, but it works.
Or maybe you have a problem and you do 方法 because you know a procedure that would solve it. But you do not 办法 because you cannot find a practical way to follow the procedure. "I know how to fix that broken door, but I cannot find the time to do it."