Consider only the mono-enol forms of the compound shown, in which one carbonyl group is converted into $\mathrm{C}=\mathrm{C}-\mathrm{OH}$ and the other stays as C=O. Counting E and Z isomers separately, how many distinct mono-enol forms are there?

The compound shown is pentane-2,4-dione. Enolisation of a carbonyl toward a terminal CH${}_3$ gives a terminal enol $\mathrm{CH}{}_2=\mathrm{C}(\mathrm{OH})\mathrm{CH}{}_2\mathrm{C}(=\mathrm{O})\mathrm{CH}{}_3$, with no E/Z isomerism. Enolisation toward the central CH${}_2$ gives $\mathrm{CH}{}_3\mathrm{C}(\mathrm{OH})=\mathrm{CHC}(=\mathrm{O})\mathrm{CH}{}_3$, which exists as E and Z isomers. Because the two carbonyls are equivalent, the total is $1+2=3$, confirmed by RDKit enumeration.