Building on Ralph's answer a bit:

Let $S,T$ be a partition of the primes into two nonempty sets (or if you prefer, the multiplicative sets generated by these); my comment on Ralph's answer was the case $T = \{p\}$.  Then the same partial fractions argument as in Ralph's answer shows that $0\to\mathbb{Z}\to S^{-1}\mathbb{Z}\oplus T^{-1}\mathbb{Z}\to\mathbb{Q}\to 0$ is an exact sequence which does not split.

With a bit more work (I'm happy to add the argument if there is interest), one can show that distinct partitions yield inequivalent exact sequences.  This gives uncountably many non-split exact sequences as Mark Grant's comment on the original post suggested there should be.