A Good Handyman is...
Skilled — This one is obvious, but the best handyman services will be offered by someone who knows what they're doing and can prove it with examples and references. Some of the skills needed are plumbing, electrical, and carpentry.
Careful — They pay attention to the details of the project from the very beginning, asking questions and examining all of the aspects of your project, no matter the size.
Equipped — You want to hire a handyman who has quality tools, a solid vehicle and can source out top of the line materials if you need them to.
Licensed - Be sure to check that they are properly licensed and insured. There have been many handyman start-ups with the turn in the economy, and they may be well qualified, however, you could get in trouble down the road if something happens or you receive an inspection by the city.
No Payment Up Front - RUN from a contractor who requires money up front. The ONLY exception is if a great amount of money is invested in materials, such as for a deck, and even then they should only ask for a minimal amount such as 30%.