In the live action version of Horrible Histories, there was a part where a Spartan family and an Athenian family have a "wife switch". In the scene when the Athenian wife came to the Spartan house, the Spartan husband told the Athenian wife to go hunt. The wife responded that Athenian women can only get out to meet other women and to go to a funeral. The husband then responded, "You are going to a funeral, the rabbit's." I realized that the response was very clever because one of the few things that the Spartans were well known for was their laconic humor (and were in fact the origin of the term "laconic"). -Counter Blitzkrieg
In 'The Borgia Family' song, there's a strange lighting effect across Lucrezia's eyes throughout the thing - which just so happens to be another 'Addams family' homage; this was how they lit Morticia in the films to make her look more eerie.
Also from the 'Borgia Family' song. Gioffre is always looking around, presumably to highlight his character as a bored 12-year-old, but then I realized—he's not just looking around, he's looking over his shoulder, checking for enemies. If you lived with that family, you'd be constantly looking over your shoulder, too.
At first, it seems odd that Gioffre isn't singing along. However, this is a reference to the fact that the boy wasn't exactly fond of the rest of his family, and is a suspect in Giovanni's murder.
Some of the musical references achieve this by matching historical ideas and figures with comparable modern day genres and artists. For example:
Charles II, a king known for his excesses and love of partying, is portrayed as a rapper.
Cleopatra with her squicky romances and love of fashion is portrayed as a Lady GagaExpy.
The women of World War Two are a girl group ("Original Girl Power!")
The Viking invaders of England are a hair metal band (the ancestors of Spinal Tap?).
The best example may be the song I'm a Greek, which extols the virtues of Ancient Greece in the style of Flanders and Swann. Donald Swann was a great admirer of Greek culture, and would no doubt heartily agree with the sentiments.
A few are much more blatant, like a song about Dick Turpin to the tune of Stand and Deliver.
The video for Crassus' Rap looks like it might be the most expensive one in the whole series - but then he is the world's richest man.