I would tend to use solid core wire for interconnections that do not have to flex. The wires will tend to stop where they are put.
Connections into and out of the board I would use multi stranded wire as they will flex more.
I am by no means a solder expert, but I tend to prefer solid wires when I'm making permanent connections between the perfboard holes. I prefer stranded wire when I'm making a cable that one end attaches to the perfboard and the other connects to things off board, since stranded works better if the cable flexes, and you use crimp connectors to make custom cables.
Thanks fungus! Exactly the kind of advice I was hoping for. AWG 30 is super thin and looks like just the right kind of wire for the project I have in mind (more about that later...). I'm going to order a pack right away. Hopefully shipping does not take ages...
You can also do a lot with the actual components. Bend the legs under the board to touch whatever it is they need to connect to but don't necessarily snip them off short.
eg. If you need some pullup resistors, put then near a chip's VCC pin. When you put in your decoupling capacitor for the chip, bend the capacitor's leg (under the board) to touch the chip's Vcc pin pin, then continue it over to where the resistors are.
If you arrange the components well this technique can really reduce the number of extra wires you need to add, especially for Vcc and GND connections.
Used wire wrap years ago. Thinking of doing it again for a few prototypes. RS used to carry it in the stores, but no longer. Have to buy online. If anyone finds it at a good price, let us know. Yes, solid wire for perfboard. I use telephone wire as it is cheap. Sometimes I can find it in the dollar store for 25ft. However I have to strip the 4 wires apart first.