Enter the cranium rat swarm (the linked monster is on Wizards’ website, but I don’t know how official it is; there are official printings of it in Fiend Folio). You’ll note it’s got a base CR of ⅛ per rat. In a swarm. The swarm gets smarter and more magical the more rats are present, and they swiftly become very dangerous, casting as a fairly-powerful sorcerer. They’re a classic D&D enemy, showing up in the excellent Planescape: Torment CRPG, and their unique mechanics make them a very interesting foe to face.
You could take this further by basing your rats on the H.I.V.E.; while the actual H.I.V.E. is a theoretical-optimization build that’s inappropriate for most games, it directs you to the Hivemind rules detailed in Book of Vile Darkness, which represent mechanics quite similar to the cranium rats. You can make very powerful foes with it.
Similar to the Verminlord build that brings the H.I.V.E. together, you might consider the various options available for a swarm druid as discussed in a previous question. A city with interesting wildlife activity seems like a great place for an evil druid to take an interest in, as a potential opportunity to forcibly “return it to nature.”
Note the uses of the word Beasts in the description of what Wild Shape does.
Beasts is a defined term; from page 2 of the Dungeon Master Basic Rules:
Beasts are nonhumanoid creatures that are a natural part of the fantasy ecology. Some of them have magical powers, but most are
unintelligent and lack any society or language. Beasts include all
varieties of ordinary animals, dinosaurs, and giant versions of
animals.
You can Wild Shape into existing "Giant" animals
For Giant (x), if the creature meet the requirements of the Beast Shape table on page 66 and the Challenge Rating requirements on the same page, then the answer is yes, you can transform into a giant beast.
For example, a Giant Scorpion is described as
Large beast, unaligned
So it meets the Beast requirements, but its CR of 3 exceeds the maximum challenge rating of 1 for an 8th level druid. So you can't Wild Shape into a Giant Scorpion.
However, a Giant Poisonous Snake is CR ¼, so you can Wild Shape into this creature if you seen it before and are at least 2nd level.
You can't Wild Shape into swarms
As for a swarm, the answer is no. This conclusion hinges on the wording of both Wild Shape and the Swarm creature.
Wild Shape (PHB page 66) opens with
Starting at 2nd level, you can use your action to magically assume the
shape of a beast that you have seen before.
The plain reading means the druid character can shape into a single beast.
The swarm descriptions are formatted like this:
(swarm size) swarm of (creature size) beasts, (alignment)
For example a Swarm of Poisonous Snakes (DM's Basic Rules, page 46) is
Medium swarm of Tiny beasts, unaligned
So it clear that the "creature" is only treated as a single entity for the purpose of combat. It is really comprised of many creatures, whose action in concert gives the swarm its power.
My opinion is that they forgot to define or haven't finalized the definition of a swarm, which is why it doesn't appear in the list of standard creature types on page 2 of the 5e Dungeon Master's Basic Rules.
Best Answer
You can, but there is little sense to do it.
There is not a rule about that, and, while the DM can do as he considers opportune, the individual creatures are often too weak to fight individually, and are no challenge at all for the PCs. Also, the Monster manual (p. 337) says the following about swarms:
So it is plausible that reducing a swarm hit points to 0 does not necessarily mean killing every one of them (unless you use a fireball or similar method). Most often it means that you killed enough of them to to make the rest of the creatures in the swarm return to their standard behaviour and flee in all directions.