Singularities of embedding operators between symmetric function spaces on $[0,1]$

The properties of the identity embedding operator $I(X_1,X_2)$, $(X_1\subset X_2)$ between symmetric function spaces on $[0,1]$ such as weak compactness, strict singularity (in two versions), and the property of being absolutely summing are examined. Banach and quasi-Banach spaces are considered. A complete description of the linear hull closed with respect to measure of a sequence $(g_n^{(r)})$ of independent symmetric equidistributed random variables with
$$ d(g_n^{(r)};t) =\operatorname{meas}\bigl(\omega: |g_n^{(r)}(\omega)|>t\bigr) =\frac 1{t^r},\qquad t\ge1,\quad 0<r<\infty, $$
is obtained, and the boundaries for this space on the scale of symmetric spaces are found.