Hello
Following a tutorial from blendercookie.com, I created water splashing in a bowl.
See attachments for renderresult, domain- and rendersettings.
The question is: how can I make the water and shadow look more smooth?
Have you set the shading to smooth for the fluid? The only thing I can think of is that each individual face is reflected something different, creating that appearene. Double check Anti-aliasing is on as well.
Other than that, I can’t help you. 
Good luck
You were right; set the shading to smooth, and turned up the anti-alising. Much better! Thank you 
Although you marked it solved it is not yet 
I can see you used a spotlight.
Spotlights are set to buffer shadows per default. You have to increase the shadow buffer size. 10240 is the maximum, 512 the minimum. It should be a power of two… 512, 1024, 2048…
Just play with the settings until the fragmentation disappears. You can also increase the samples. However:
If you change to raytraced shadows it will be all smooth but much slower.
But then you can also set the materials to “receive transparent shadow” this way the shadow of the water will not be all hard like the water has no opacity at all.