Hair bleach, which is mixed with an aqueous oxidant formulation just before use, is a suspension containing (a) 0.1-80 wt.% organic lipophilic compound(s) (I), (b) 0.1-40 wt.% lipophilic (in)organic thickener(s), forming an oleogel or lipogel with (I), (c) 0.1-40 wt.% hydrophilic (in)organic thickener(s), (d) 10-65 wt.% inorganic per-salt(s), (e) 10-45 wt.% alkaline salt(s) and optionally ancillaries and additives. Hair bleach, which is mixed with an aqueous oxidant formulation just before use, is in the form of a suspension containing (a) 0.1-80 wt.% organic lipophilic compound(s) (I) selected from vegetable and animal fats, oils and waxes, paraffin hydrocarbons, higher alcohols and ethers, aliphatic and aromatic esters and silicone oils, (b) 0.1-40 wt.% lipophilic (in)organic thickener(s) (II), forming an oleogel or lipogel with (I), (c) 0.1-40 wt.% hydrophilic (in)organic thickener(s) (III), (d) 10-65 wt.% inorganic per-salt(s), (e) 10-45 wt.% alkaline salt(s) and optionally ancillaries and additives.