Morning Menta,

The 877A is a bit tempremental at times. I use it for a lot of projects, and I find that some times it works under strange conditions when it shouldn't. We've established that you have a resistor between VDD and MCLR, try wiring the MCLR direct to the 5V line. You will need 18pF or 22pF caps on that crystal. Back to super short leads and the caps between the xtal and ground on both sides. It's not just because we say it, it's because thats how they work.....
Your transformer (I noticed you never answered what the rating is/was) probably has a lot of noise on it, so try putting a 470uF cap accross the output of the tranny and if possible, accross the output of the 7805. That will smooth out any ripples that may be around. Just because it SAYS it has a complete DC output doesn't mean it's true.
Hope that helps some