The sound drivers are not included in the dCore kernels but can be imported through the dCore extension alsa-modules-<kernel>-tinycore. The dCore sound README page about ALSA provides a complete walkthrough.

• The Tiny Core wiki contains several articles about setting up sound, two (one and another one) about ALSA plus one about OSS and one about PulseAudio.
• Sound management is a complicated issue. This can be seen from the fact that just about every distribution gives it ample consideration in their wiki. The Archlinux-wiki, for instance, is an excellent source of information.
• A short note to newcomers, if everything is set up well but sound does not work ensure the appropriate channels are not muted. For example, run alsamixer, ensure the volume bars are turned up and no applicable channels are muted (designated with (mm) below volume slider).

Alsamixer, for instance, will show the volume bars turned up but mark a muted channel by a letter M below it. This is easy to miss (trust me, or, if not, cf. screen shot).