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The behaviour of sulfur in magmas is complex because it dissolves as both sulfide (S 2- ) and sulfate (S 6+ ) in silicate melt. An interesting aspect in the behaviour of sulfur is the solubility minima (SS min ) and maxima (SS max ) with varying oxygen fugacity ( f O 2 ). We use a simple ternary model (silicate–S 2 –O 2 ) to explore the varying f O 2 paths where these phenomena occur. Both SS min and SS max occur when S 2- and S 6+ are present in the silicate melt in similar quantities due to the differing solubility mechanisms of these species. At constant T , a minimum in dissolved total S content in vapour-saturated silicate melt ( w S T m ) occurs along paths of increasing f O 2 and either constant f S 2 or P . For paths on which w S T m is held constant with increasing f O 2 , the SS min is expressed as a maximum in P . The SS min occurs when the fraction of S 6+ in the melt ([S 6+ /S T ] m ) is 0.25 for constant f S 2 and [S 6+ /S T ] m ≅ 0.75 for constant w S T m and P . A minimum in w S T m is not encountered during closed- or open-system depressurisation in the simple system we modelled. However, the SS min marks a change from reduction to oxidation during degassing. Various SS max occur when the silicate melt is multiply-saturated with at least two phases: vapour, sulfide melt, and/or anhydrite. The SS min and SS max are important features of magmatic process involving S, such as mantle melting, magma mixing, and degassing. These concepts influence calculations of the pressures of vapour-saturation, f O 2 , and SO 2 emissions using melt inclusions. Supplementary material: Additional information and data used to create the figures are available at https://doi.org/10.6084/m9.figshare.c.6274527 . The code used to generate the data is available at https://github.com/eryhughes/SSminmax . Thematic collection: This article is part of the Sulfur in the Earth system collection available at: https://www.lyellcollection.org/topic/collections/sulfur-in-the-earth-system