Get a set of JIS screwdrivers. Do a search on this site, I think I bought mine from McMaster Carr after seeing a link on here, possibly by Igofar. Nothing fits a Honda phillips screw better, and it makes an amazing difference in loosening tight screws.
The screws aren't specific to Honda; they're defined by Japanese Industrial Standard B 1012 and can be found on most things made in Japan.
The Phillips design is set up so that if the screw stalls (reaches the right torque), the driver will cam out or, if forced, the driver will be damaged and the screw head will be left intact. (Doesn't always work out that way if the screws are made of cheap metal.) This was on purpose, because when the Phillips screw head was designed in the 1930s, there weren't good tools to limit torque. Unfortunately, they didn't think the design out very well and the screwdriver cams out in
both directions, which can make removing overtorqued screws difficult.
The JIS design came about after World War II and doesn't cam out. wBW's
Vessel screwdriver review has a couple of good pictures that illustrate the differences. Don't try too hard to remove an overtorqued JIS screw with a Phillips driver; odds are good you'll damage both.
If you have an older ST that came with a tool kit, the screwdriver in that is JIS. For STs built during the WonderFoam era, the screwdriver is available from Honda parts.
--Mark