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I created all the 3 faces attached to the extruded cube with shift+F. I tested some variations and the results were prettym uch the same. It uses some edges to make a nice smooth subdivision and it doesn't use others as subdivision orientation. There, it's sharp. Why is that? How can I make it round where it's sharp? This is with faces inside. This time, it worked somehow. There's still a lot of sharpness: enter image description here When the inside faces are removed, it spikes again. There are never zero value faces (shown): enter image description here

The file:

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    $\begingroup$ Vertices aren't being included or excluded from Subsurf modifier influence; if it creates artifacts then there's something wrong with geometry. Try removing doubles, see if there any zero-area faces (like extruded and left in place face) etc. $\endgroup$ – Mr Zak Feb 16 at 20:58
  • $\begingroup$ I've edited my question and shared our progress. It's still not as I wish. $\endgroup$ – Knampf Feb 16 at 21:17
  • $\begingroup$ It looks like you have non-manifold geometry, which is going to cause you endless problems. This is not the way to go about making sharp edges with subdivision. You need to add supporting edges to either side of the edges that you want to harden, apply a bevel modifier and weight those edges. It is hard to say more without seeing the blend file, so upload it if you can. $\endgroup$ – Rekov Feb 16 at 21:26
  • $\begingroup$ I've put the file in. $\endgroup$ – Knampf Feb 16 at 22:11
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Your biggest problem are these internal faces. I can't really think of a reason you would ever want something like this, so start by deleting them.

The remaining problem, faces have fronts and backs, and the reason you're getting that cusp is because some of your faces are pointing the wrong way.

Select the whole mesh with 'A' and hit 'Shift N' to automatically correct the surface normals.

enter image description here

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