I assume that all manifolds are closed and connected (embedding in $\mathbb R^n$ or in $S^n$ is the same), but not necessarily orientable.

Concerning dimension 3, a famous [theorem of Wall](http://www.ams.org/mathscinet/search/publdoc.html?r=1&pg1=CNO&s1=175139&loc=fromrevtext) states that every closed 3-manifold embeds in $S^5$.

It is not possible to improve this result. A [theorem of Shiomi][1] shows that there is no closed 4-manifold which contains every possible closed 3-manifold.

The question in dimension 4 seems more complicated. Since $4=2^2$, this is one of the dimensions where the real projective space $\mathbb R\mathbb P^4$ embeds in $S^8$ and not in $S^7$. Every <i>orientable</i> 4-manifold embeds <i> topologically </i> in $S^7$ by a [theorem of Fuquan Fang](http://www.ams.org/mathscinet/search/publdoc.html?amp=&loc=refcit&refcit=1286925&vfpref=html&r=3&mx-pid=1923991). 

 [1]: https://projecteuclid.org/journals/osaka-journal-of-mathematics/volume-28/issue-3/On-imbedding-3-manifolds-into-4-manifolds/ojm/1200783230.full