  We give lower and upper bounds for the separation profile (introduced by
Benjamini, Schramm & Tim\'ar) for various graphs using the isoperimetric
profile, growth and Hilbertian compression. For graphs which have polynomial
isoperimetry and growth, we show that the separation profile $\mathrm{Sep}(n)$
is also bounded by powers of $n$. For many amenable groups, we show a lower
bound in $n/ \log(n)^a$ and, for any group which has a non-trivial compression
exponent in an $L^p$-space, an upper bound in $n/ \log(n)^b$. We show that
solvable groups of exponential growth cannot have a separation profile bounded
above by a sublinear power function. In an appendix, we introduce the notion of
local separation, with applications for percolation clusters of $
\mathbb{Z}^{d} $ and graphs which have polynomial isoperimetry and growth.
