Can I use the 10 gauge wires (red, black, white, ground)from 30 amp, 250 volt dryer receptacle for two 125 volt circuits for two separate duplex receptacles? Can I use stranded wire for jumpers, or must I use solid wire?
You could make a MWBC out of this /3 + gnd cable and install two 20 A receptacles. You would change the 2-pole 30 A breaker to a 2-pole 20 A breaker.
If the conductors are aluminum, you would pigtail with 12 AWG copper. If the conductors are 10 AWG copper, you might be able to connect these directly to the contacts on the receptacles.
I would make these to 20 A so that if you ever needed to run heavy current demand appliances, you'd have the capability.
EDIT To install a single duplex receptacle in the box, you would split the hot sides by breaking the connecting tab. The black wire would go to the hot side of one receptacle and the red to the hot side of the other.
The neutral side tab would be left intact.
Do you really need a split outlet? Why not just cap the red wire and use the black, white and ground. You have a 30 amp breaker that should be changed out to a 15 or 20 depending on the rating of the receptacle you're installing. Disconnect and cap the red wire in the panel, leave the breaker there so there are no open spaces in the panel. Install the new breaker in an available slot.
What you are trying to do can already be accomplished by existing adaptors. Here's an example converting dryer plug to split receptacle: