How do I dig a 3 foot deep narrow trench in soft clay
It's physically very difficult to dig a very narrow but deep trench. You may have better luck just renting a ditch digger/trencher, which is kind of like a big chainsaw for dirt. You can dig trenches that are very narrow, and the only thing that really slows you down with it is large rocks. 



You should have utilities marked before you dig (whether using a trencher or not). In some areas, this is enforced by law (eg: in Ontario, Canada, it is law under the Ontario Occupational Health And Safety Act).