If memory serves me correctly, all alkali metal salts are soluble.
It depends on anion. KClO4, K2[PtCl6 ], K2Na[Co(NO2)6 ] are poor soluble, Na[Sb(OH)6 ], Li2CO3,
Rb2Na[Co(NO2)6 ], Cs2Na[Co(NO2)6 ] also. Poor solubility means in this case less than 1g/L, sometimes a few mg/L.