I will answer for embeddings of Sⁿ in Sⁿ⁺³.

No in the PL category by Zeeman.
No in the topological category by Stallings.
Yes in the differentiable category. See Levine and Haefliger.

I will answer for embeddings of Sⁿ in Sⁿ⁺³.

No in the PL category by Zeeman.
No in the topological category by Stallings.
Yes in the differentiable category. See Levine and Haefliger. 

EDIT: In the case $n=2$, which the question is about, the result is no in the differentiable category as well, as pointed out in other answers.