Why do the L or R windshield washers sometimes fail--but never at the same time?

SOURCE: Windshield wiper operation

I have seen a simular fault which ended up being the intermittent wiper relay was full of water from water collected in the airconditoners evaporator tray.
The water has problems escaping this tray & when you go around the corners the water rans out onto your feet & the intermittent wiper relay is mounted down the side of the transmission wall & gets soaked along with the carpet.
I suggest you find the relay, remove it check the inside of it for water damage. If stuffed replace it & also suggest to check your evaporator's tray & rubber S bend drain for obstructions. I think this is a design fault in all 240s.
Another fault I have seen with the wiper motor is it not running too well with excessive noise, due to water getting into the motor through the 2 mounting clips on the round part of the motor casing to the casting. There are 2 small slots which allows water into the motor.
I suggest remove the wiper motor by removing the cir clip & or nut under the dash board connecting to the arms, Then the 3 x 10mm bolts & carefully pull the wiper motor apart don't loose the 2 angled clips & screws, clean & re grease the motor & try not to damage the maganets or over tighten the 2 screws & re assemble in reverse order.

SOURCE: The windshield wipers and washer fluid dispenser stay on

This must be due to a defective windshield-washer-switch. Repacement of it is the only way.

SOURCE: Front windshield washers do not operate?

It could be the motor or the windshield washer bars. The bearings could have rotted off. Hope this helps!?

Please rate solution - Thanks.

SOURCE: My Mom's 1989 Volvo 240 front wipers work one day

Bad electrical contact, to be expected in such an old car. You will need to check the command cluster on the steering wheel shaft (the switch might have failed), the relay (located to the left of the clutch pedal, behind the insulation and a small metal plate - just pull the wires out from the top, and you should find it) and the contact to the motor itself.
If all are ok - power is reaching the motor but the motor doesn't move - then the wiper motor will have to be replaced (top left in the engine bay). Again, something to be expected on an old car. So identify the failed part and replace it. Here are the new parts
http://www.partsgeek.com/catalog/1989/volvo/244/body_electrical/wiper_switch.html
http://www.partsgeek.com/catalog/1989/volvo/244/body_electrical/wiper_relay.html
http://www.partsgeek.com/catalog/1989/volvo/244/body_electrical/windshield_wiper_motor.html

SOURCE: Windshield wipers working intermittently. Sometimes

This is the wiper motor failing...But to be safe unplug wiper motor,turn on wiper switch,connect test light to negative side of battery or good solid metal ground and stick probe in connector.if it is not working and the test light is not on then the wiper switch has failed.....and if the test light comes on at connector the wiper motor has failed..Since it works intermittently have an assistant to work the wiper switch and test a number of times to be sure.You can buy a cheap test light that will be good enough to check this at any auto parts store...BEFORE YOU BEGIN YOUR TEST MAKE SURE TEST LIGHT
HAS GOOD GROUND BY CLAMPING TO NEGATIVE SIDE OF BATTERY AND TOUCH PROBE TO POSITIVE SIDE OF BATTERY TO ENSURE GOOD CONNECTION...LIGHT WILL COME ON.