I am just learning Morse theory. I apologize if the solution to the following question is well-known.
For a Morse function $f(x):\mathbb{R}^n\mapsto\mathbb{R}$, if the level set $f^{-1}(c)$ only contains regular points, is $f^{-1}(c)$ always diffeomorphic to a collection of $(n-1)$-spheres? If no, what conditions can make this happen?
No, for example if $f(x) = \langle a,x\rangle$ with $a\in \mathbb{R}^n\setminus\{O\}$ then $\nabla f(x) = a \neq 0$ so that any point is regular but the fiber $f^{-1}(y)$ is an affine hyperplane.
What you can say is that if $f^{-1}(c_0) \simeq \mathbb{S}^{n-1}$ for some $c_0\in \mathbb{R}$ and $f$ has no critical points ($f$ smooth). Then the inverse image of any point $c_1$ such that $f^{-1}([c_0,c_1])$ is compact is diffeomorphic to $\mathbb{S}^{n-1}$ (it is also smoothly isotopic). So for example if $f$ is proper then the fiber of any point in the image will be diffeomerphic to $f^{-1}(c_0)$.