The spectral analysis of Aharonov-Bohm Hamiltonians with flux $\frac12$ leads surprisingly to a new insight on some questions of isospectrality appearing for example in [JLNP, LPP] and of minimal partitions [HHOT]. We will illustrate this point of view by discussing the question of spectral minimal $3$-partitions for the rectangle $]-\frac a2,\frac a2[\times ] -\frac b2,\frac b2[\,$, with $0< a\leq b$. It has been observed in [HHOT] that when $0<\frac ab < \sqrt{\frac 38}$ the minimal $3$-partition is obtained by the three nodal domains of the third eigenfunction corresponding to the three rectangles $]-\frac a2,\frac a2[\times ] -\frac b2,-\frac b6[$, $]-\frac a2,\frac a2[\times ] -\frac b6,\frac b6[$ and $]-\frac a2,\frac a2[\times ] \frac b6, \frac b2[$. We will describe a possible mechanism of transition for increasing $\frac ab$ between these nodal minimal $3$-partitions and non nodal minimal $3$-partitions at the value $ \sqrt{\frac 38}$ and discuss the existence of symmetric candidates for giving minimal $3$-partitions when $ \sqrt{\frac 38}<\frac ab \leq 1$. Numerical analysis leads very naturally to nice questions of isospectrality which are solved by introduction of Aharonov-Bohm Hamiltonians or by going on the double covering of the punctured rectangle.