Which of the following is an anionic detergent?

First, let us recall the basic classification of synthetic detergents. Detergents are broadly placed into three categories based on the electrical charge that appears on the part of the molecule which goes into water when the detergent is dissolved.

We have

$$\text{1. Anionic detergents}$$ — their hydrophilic end carries a negative charge (usually $$\;{\large -SO_3^-}\;$$ or $$\;{\large -SO_4^-}\;$$).
$$\text{2. Cationic detergents}$$ — their hydrophilic end carries a positive charge (most commonly the quaternary ammonium ion $$\;{\large R_4N^+}\;$$).
$$\text{3. Non-ionic detergents}$$ — their hydrophilic end is neutral (generally an alcohol, ester, or ether chain).

Now we shall look at each option one by one and identify the nature of the hydrophilic (water-loving) part.

Option A : Cetyltrimethyl ammonium bromide
The name shows a quaternary ammonium group $$\;{\large (CH_3)_3N^+}$$ with a bromide counter-ion $$\;Br^-$$. Because the positive charge resides on nitrogen, the detergent ion produced in water is $$\;{\large (CH_3)_3N^+R}\;$$. A detergent whose head bears a positive charge is, by definition, cationic, not anionic.

Option B : Glyceryl oleate
This substance is an ester formed from glycerol and oleic acid. No ionic group is present; the molecule remains neutral in aqueous solution. Therefore it belongs to the non-ionic class, not to the anionic class.

Option C : Hexadecyltrimethyl ammonium bromide
Again we notice the quaternary ammonium ion $$\;{\large (CH_3)_3N^+}$$ paired with bromide. Exactly like option A, it is a cationic detergent.

Option D : Sodium lauryl sulphate
The structural formula of lauryl sulphate ion may be written as

$$\text{Lauryl group} - SO_4^- \;=\; CH_3(CH_2)_{11}OSO_3^-$$

In aqueous solution the compound dissociates as

$$\underbrace{CH_3(CH_2)_{11}OSO_3^-}_{\text{detergent ion carrying a negative charge}} \;+\; \underbrace{Na^+}_{\text{sodium counter-ion}}$$

The hydrophilic end clearly possesses a negative charge (the sulphate group $$-SO_4^-$$). Hence sodium lauryl sulphate is an anionic detergent.

Collecting our observations, only option D furnishes an ion with a negative charge in solution, satisfying the definition of an anionic detergent. Therefore the correct choice is option D.

Hence, the correct answer is Option D.