So in the lore, Riven lost a battle and when she called for backup, Singed tossed in chemicals killing Noxian and Ionians alike. My question is why did she lose the battle in the first place if she had been training for so long?

If you're looking for the whole story, Youso's source is Riven's League Judgement. It's pretty awesome

she could do something...but singed finished them all alike

Riven's company was travelling through a valley where, to the best of their estimations, the enemy had not only suffered a notable defeat that morning but had barely even managed to get together half of the forces the Noxians had expected. All indication was that her column wasn't going to face any sort of resistance, much less that which might pose a real threat.

We have no idea what scale the battle that preceded her arrival might have been, nor how many Ionians were waiting in ambush. Assuming we go off the typical modern definition of "company", and ignoring any potential losses the campaign might have brought, she could've been leading as many as 100 men. No matter how capable a soldier she might have been, I really can't see her doing anything to offset the sort of numbers advantage the Ionians must've had.

Think of it sorta like when the Romans went into England
They may be trained, but they are outnumbered

She chased singed.