I know that retiarius fought with tridents and nets in gladiatorial combat, but have there been other significant examples of spears being used with sidearms other than shields?
There a few instances of warriors using spears one-handed without a shield, though I cannot think of any that used them alongside another side-arm.
The one example that comes to mind is that of Achaemenid cavalry. The Achaemenids were a dynasty of Persian origin that ruled much of the Near East from the 6th through to the 4th century BC. This image from a sarcophagus found in Anatolia dating from the Achaemenid period shows a cavalryman attacking an opponent. They are holding the reigns of the horse in one hand, and their spear in another:
Similar representations are found on Achaemenid seals. There is also references by a Greek writer, Herodotus, to Persian cavalry using javelins:
'and when the horsemen had ridden to attack them, they did damage to the whole army of the Hellenes by hurling javelins against them and shooting with bows, being mounted archers and hard therefore to fight against'
The use of javelins appears to have been very common. Another writer, Xenophon, describes a Persian noble, Cyrus the Younger, in the following way:
'Cyrus sprang from his carriage and donned his corselet; then leaping on to his charger's back, with the javelins firmly clutched, he passed the order to the rest, to arm themselves and fall into their several ranks.'
Note that there is no reference to Cyrus using a shield. So if we took all this evidence together it is plausible to imagine horsemen would first throw missiles at an opponent before moving to close combat using a spear in one hand, with a shield being completely absent.
Sources:
The Anabasis, by Xenophon: https://www.gutenberg.org/files/1170/1170-h/1170-h.htm
The History of Herodotus, Volume 2: https://www.gutenberg.org/files/2456/2456-h/2456-h.htm
