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Additives which are ALWAYS of animal origin, such as (HARAAM):

E120 Cochineal : a red colour obtained from female insects

E441 Gelatine : derived from the bones and/ or hides of cattle and/ or pigs

E542 Edible Bone Phosphate : an extract from animal bones

E904 Shellac : a resin from the lac insect

Whilst some additives with a common code such as E47, can be either of animal or plant origin and this latter type needs to be investigated on a case-by-case basis per product/ manufacturer.
The main additives you need to be aware of are:

Glycerol / Glycerin / Glycerine (E422) - haraam if obtained from pork or non-halal meat sources.
Emulsifiers (E470 to E483) - haraam if obtained from pork or non-halal sources.
Edible Bone Phosphate (E542) - haraam if obtained from pork or non-halal meat sources.

E100
Curcumin/Turmeric
Halal if pure grind turmeric powder or granular. Haraam if hidden ingredient is pork fat based emulsifier in dry mix.

E101
Riboflavin (Vitamin B2)
Mushbooh (Haraam if from pork liver & Kidney, Halal if 100% plant material

E120
Cochineal / Carminic Acid
Haraam according to Hanafi Madhab

E132
Indigo Carmine / Idigotine
Color
Halal if use as is as a 100% synthetic color but if pork glycerin is added as a solvent then it is Haraam

Beta-apo-8-carotenal
Halal if used as 100% dry powder or granular or Halal if vegetable oil is used as a solvent in liquid form as a hidden ingredient. Haraam if pork gelatin is used as hidden ingredient or carrier

E160f
Ethyl ester of Beta-apo-8-cartonoic acid
Halal if used as 100% dry powder or granular or Halal if vegetable oil is used as a solvent in liquid form as a hidden ingredient. Haraam if pork gelatin is used as hidden ingredient or carrier
E304
Ascorbyl Palmitate
Antioxidants- Vitamin C and derivatives
Halal if saturated fatty acid Palmitic acid is obtained from plant. Haraam if palmitic acid is obtained from pork fat

E306
Natural Extracts rich in Tocopherols
Antioxidants- Vitamin E
Halal If Tocopherol is obtained from plant fat. Haraam if Tocopherol is obtained from pork fat

E307
Synthetic Alpha-Tocopherol
Antioxidants- Vitamin E
Halal only if it is made with all Halal synthetic material without alcoholic fermentation synthetic method

E308
Synthetic Gamma-Tocopherol
Antioxidants- Vitamin E
Halal only if it is made with all Halal synthetic material without alcoholic fermentation synthetic method

E309
Synthetic Delta-Tocopherol
Antioxidants- Vitamin E
Halal only if it is made with all Halal synthetic material without alcoholic fermentation synthetic method

E320
Butylated Hydroxyanisole (BHA)
Antioxidants- other
Halal if only vegetable oil is used as a carrier. Haraam if the carrier is from pork fat. It is not available as pure 100% chemical.

E321
Butylated Hydroxytoluene (BHT)
Antioxidants- other
Halal if only vegetable oil is used as a carrier. Haraam if the carrier is from pork fat. It is not available as pure 100% chemical.

E322
Lecithin
Emulsifiers and Stabilizers
Halal if obtained from soy fat or egg yolk in Europe. It is Halal in USA because it is always obtained from soy fat.