Strictly increasing but not convex preferences
Take $u(x,y)=x^2+y^2$ on $\mathbb R_+^2$. The function is strictly increasing in both $x$ and $y$, but the indifference curves are concave to the origin. Hence the preference it represents cannot be convex. For example, $u(1,0)=u(0,1)=1$, but the average of the two bundles is less preferred:
\begin{equation}
u\left(\frac12,\frac12\right)=\frac12&lt;1=\frac12u(1,0)+\frac12u(0,1).
\end{equation}
The indifference curves for perfect substitutes are straight lines. They still represent a convex preference, although not a strictly convex one.