I have a cube and by subdividing faces, now I have a bunch of vertices in each face.
How can I arrange some selected vertices in a curved shape (they are now in a straight line shape)?
Then I want to extrude those curved shape vertices.
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Another option is:
removing the row of vertices you want to make curved
adding a Bezier curve in its place.
"Resolution Preview U" should be set to exactly $N-1$ where $N$ is the number of vertices making a side of your cube.
Also, you might want to make sure that the Bezier curve lays completely on the XY plane
converting the curve to mesh
joining the curve and the cube
reconstructing the faces by selecting the two adjacent rows and doing "Bridge Edge Loops" (must be done once per side)
You can enable "Loop tools" add-on, and use its relax, curve commands for example (loop tools menu is accessible with W key menu). Or you can use the proportional editing mode ("O" key) to move verts smoothly. Also smooth command (W menu) may be handy to round some edges.
Tip: to isolate selected verts for proportional editing, just hide all other verts by (Shift + H). Then after editing, unhide them back (Alt + H).