The description of the Weave of Magic in the Spellcasting chapter says Bards are users of Arcane magic, but they seem to only be offered the use of an instrument as a focus. Can they use a standard Arcane focus?

Bards can't use an Arcane Focus

In the Bard class features there is a "Spellcasting Focus" section stating:

You can use a musical instrument (see the Tools section) as a spellcasting focus for your bard spells.

Further, the description of an Arcane Focus states:

A sorcerer, warlock, or wizard can use such an item as a spellcasting focus.

Since the bard is not included in the list of classes in the Arcane Focus description and the Spellcasting Focus section only mentions musical instruments, bards can't use a normal Arcane Focus.

As pointed out by David Coffron, this is also true for Arcane Tricksters and Eldritch Knights despite them using Arcane magic because they do not have an entry for using a spellcasting focus in their description (and Arcane Focus does not mention them)

Bards cannot use an arcane focus

The description of the Arcane Focus which explicitly defines which classes are allowed to use it as a focus:

A sorcerer, warlock, or wizard can use such an item as a spellcasting focus.

No bards are allowed it seems.

Foci are determined by class, not spellcasting type

It seems that the type of focus allowed to be used is defined by the class and not by what type of caster they are.

The Bard class says:

You can use a musical instrument [...] as a spellcasting focus for your bard spells.

The wizard class says:

You can use an arcane focus [...] as a spellcasting focus for your wizard spells.

Both are "arcane" classes. But the class specifies what can be used. If an arcane focus was allowed to be used by all arcane casters, then the wizard wouldn't specify it as their only option.

For what it is worth, Arcane Tricksters and Eldritch Knights also cannot use an arcane focus despite also being arcane casters for the same reasons. (Thanks @DavidCoffron)

Jeremy Crawford also indirectly supports this in an unofficial tweet:

Every spellcaster—including bards—can use a component pouch for spellcasting.

Some classes also know how to use certain objects as a spellcasting
focus. Bards, for example, can use musical instruments in this way,
but they aren't required to.

^{It just goes to show that arcane foci are very, very classy.}