The list of optimal systolic inequalities appearing below is surely incomplete. Please signal omissions to katzmik (at) macs.removethis.biu.andthis.ac.il

Optimal inequalities for closed surfaces:

1.
Loewner inequality
for the torus ('49)

2.
Pu's inequality
for the real projective plane ('52)

3.
Rodin's optimal inequality
for closed surfaces ('68)

4.
Bavard's inequality
for the Klein bottle ('86)

5. In genus 2:
F. Jenni ,
C. Bavard,
P. Schmutz
in hyperbolic case. S. Sabourau and M. Katz in CAT(0) case

6. For relative systole of the quotient of a hyperelliptic surface by
a fixed point free, real involution, by V. Bangert, C. Croke,
S. Ivanov, and M. Katz

7. Relative systole of genus 2 surface, in hyperbolic case:

H. Parlier

Optimal inequalities for manifolds of dimension 3 or more:

9. Gromov's inequality for the
stable 2-systole
of complex projective space

10. Burago-Ivanov-Gromov inequality for the stable 1-systole of the
n-torus (or more generally, n-manifold with real cup-length n), in
terms of the n-th Hermite constant

11. An optimal (1,n-1) inequality, in terms of the n-th
Bergé-Martinet constant, by Bangert and Katz

12. inequality combining 1-systole and stable 1-systole, in terms of
the Hermite constant, generalizing item 10, by Ivanov and Katz

13. Inequality of item 12 is valid for 3-manifolds with
b_{1}=2 and with non-vanishing Lescop invariant, by Katz and
Lescop based on argument of Marin

14. Inequality combining 1-systole and conformal 1-systole,
generalizing the above, by Bangert, Croke, Ivanov, Katz

Most of the above inequalities are discussed in the recent monograph
Systolic geometry and topology.