We present algorithms for variants of the 3XOR problem with lists consisting of random sparse $n$-bit vectors. We consider two notions of sparsity: low-density (each bit is independently biased towards zero) and low-weight (the Hamming weight of $n$-bit vectors is fixed). We show that the random sparse 3XOR problem can be solved in strongly subquadratic time, namely less than $\bigO{N^{2-\epsilon}}$ operations for a constant $\epsilon > 0$. This stands in strong contrast with the regular case, where it has not been possible to have the exponent drop below $2 - o(1)$. In the low-density setting, a very simple algorithm even runs in linear time with overwhelming success probability when the density is less than $0.0957$. Our algorithms exploit the randomness (and sparsity) of the input in an essential way.