Electrical resistivity, magnetic susceptibility, and specific heat measurements on single crystals of La$Tr_{2}$Al$_{20}$ ($Tr$ = Ti, V, Nb, and Ta) revealed that these four compounds exhibit weak-coupling superconductivity with transition temperatures $T_{rm c}$ = 0.46, 0.15, 1.05, and 1.03 K, respectively. LaTi$_{2}$Al$_{20}$ is most probably a type-I superconductor, which is quite rare among intermetallic compounds. Single-crystal X-ray diffraction suggests rattling anharmonic large-amplitude oscillations of Al ions (16$c$ site) on the Al$_{16}$ cage, while no such feature is suggested for the cage-center La ion. Using a parameter $d_{rm GFS}$ quantifying the guest free space of the cage-center ion, we demonstrate that nonmagnetic $RTr_{2}$Al$_{20}$ superconductors are classified into two groups, i.e., (A) $d_{rm GFS} e 0$ and $T_{rm c}$ correlates with $d_{rm GFS}$, and (B) $d_{rm GFS} simeq 0$ and $T_{rm c}$ seems to be governed by other factors.
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