Let $Y$ be a stochastic process on $[0,1]$ satisfying $\rm dY(t)=n^{1/2}f(t)\rm dt + \rm dW(t)$, where $n\ge 1$ is a given scale parameter (`sample size'), $W$ is standard Brownian motion and $f$ is an unknown function. Utilizing suitable multiscale tests, we construct confidence bands for $f$ with guaranteed given coverage probability, assuming that $f$ is isotonic or convex. These confidence bands are computationally feasible and shown to be asymptotically sharp optimal in an appropriate sense.