Ik leg het mes in de lade.

All four verbs can mean “to put” or “to place,” but they have slight nuances:

• leggen literally means “to lay” something down in a horizontal position.
• zetten implies “to set” something upright or in a standing position.
• plaatsen is more neutral/formal, “to place” or “position.”
• stoppen suggests “to stuff” or “to stick” something into an opening.

In this context, you’re placing a knife (horizontal) into a drawer, so leggen is the most natural. You could say Ik stop het mes in de lade if you imagine pushing it in among other items, but leggen is by far the most common choice here.

leggen is an action verb (transitive): you actively place something. It takes a direct object: Ik leg het mes.
liggen is a state verb (intransitive): something “lies” somewhere. Example: Het mes ligt in de lade (“The knife is in the drawer”).

Use leggen when you put it there; use liggen to describe where it already is.

Dutch has two grammatical genders: common (de) and neuter (het). There’s no simple rule, so you have to learn each noun’s gender. A few hints:
• Many diminutives (like tafeltje) are neuter (het tafeltje).
• Nouns ending in –ing, –heid, –ie are common.
• Mes is irregularly neuter, so it takes het.
• Lade is a common noun, so it takes de.

Always check a dictionary when in doubt.

Dutch main clauses follow the “Verb-second” (V2) rule. The finite verb (leg) must be the second element. Standard order: Subject – Verb – Object – Adverbials (including prepositional phrases). So:

1. Ik (subject)
2. leg (verb)
3. het mes (object)
4. in de lade (location)

You could front the location for emphasis: In de lade leg ik het mes, but then the verb stays second and the subject follows it.

Dutch often uses the present perfect for past events. You have two options:

1. Imperfect (simple past): Ik legde het mes in de lade.
2. Present perfect (more common in speech): Ik heb het mes in de lade gelegd.

Note that leggen is irregular in the perfect: the past participle is gelegd.

Leggen here takes two complements:

1. A direct object (what is being laid): het mes
2. A prepositional object indicating location: in de lade

So you have one direct object and one prepositional phrase.

• in de lade means “inside the drawer.”
• op de lade means “on top of the drawer.”

Choose in when you’re placing the knife into the compartment; use op if you leave it on the outside surface.