I too have these shoots at the bottom of my Yucca's. I was told it was a response to the stressful winter we and the poor plants have endured.

Yucca or Cordyline? Yucca filamentosa and Yucca gloriosa (the two commonest species) are pretty tough and wouldn't have suffered much during this winter, mine didn't.

Cordyline is different though and many have been damaged with the resuly often being sprouting shoots from the base.