Purely unrectifiable sets, fractal percolation and graphs of functions

This paper contains a survey of some of the results of the author related to unrectifiablity and is an extended version of the author’s talk given at the Second Winter School Geometric Measure Theory Rectifiability vs. Pure Unrectifiability in Hanghzou, China. These results include irregular/purely unrectifiable $1$-sets on the graphs of continuous functions like the Takagi, the Weierstrass-Cellerier and the typical (in the sense of Baire) continuous function. It is also discussed that there exists ${\alpha }_0<1$ such that the fractal percolation is almost surely purely $\alpha$-unrectifiable for all $\alpha >{\alpha }_0$. The background of the $1$-unrectifiability is discussed in more detail.