Learn Dutch: Laten we samen de spandoeken vasthouden.

Laten we is used to make a suggestion, just like "Let's" in English. In this structure, laten acts as the conjugated verb, and it pushes the main action verb (vasthouden) to the very end of the sentence in its full dictionary form (the infinitive).

Is vasthouden a separable verb? Why isn't it split up?

Yes, vasthouden (to hold / to hold on to) is a separable verb made of vast (fixed/secure) and houden (to hold). However, because laten we requires the full infinitive at the end of the sentence, the verb stays glued together as one word here.

Why does samen come before de spandoeken?

In Dutch, short adverbs like samen (together) usually come early in the sentence, right after the subject and the conjugated verb. The direct object (de spandoeken) follows it, and the infinitive verb must take the very last spot.

How do you form the singular of de spandoeken? Is it a "de" or "het" word?

The singular is het spandoek. It is a compound word ending in het doek (the cloth or canvas), so it takes the neuter article het. All plural nouns use de, which is why it becomes de spandoeken here.