Arthur Poe and his sons Edgar and Albert are named after gothic writer Edgar Allan Poe. Edgar and Albert may also be named after poet Edgar Albert Guest. Eleanora Poe is named after "Eleonora" by Edgar Allan Poe.
The yellow trolley passing through the gray Briny Beach, might be a reference to Gatsby's yellow car driving through the Valley of the Ashes in The Great Gatsby by F. Scott Fitzgerald. A Series of Unfortunate Events has a lot of Great Gatsby references.
The trolley driver asks the Baudelaires if they're going to the fair. The silhouette of a Ferris wheel from the fair is later shown in the distance when the Baudelaires are on the trolley with Justice Strauss.
The dark red goblet Count Olaf holds in the "Pasta Putanesca" scene is from the "1876 Cape Cod Selection. Designed and sold by Avon, the dinnerware collection has 37 pieces in total and was manufactured by Wheaton Glass from 1975 until production ceased in 1993. The line was officially discontinued in 1995.
The trivia items below may give away important plot points.
As Lemony is walking through the sewers there are signs with names and locations which foreshadow characters and events in the future. These include: Montgomery (Uncle Monty, The Reptile Room); Anwhistle (Aunt Josephine, The Wide Window); Orwell (Georgina Orwell, The Miserable Mill); Quagmire (The Quagmire Triplets, The Austere Academy); 667 (the address the Baudelaire's move to in The Ersatz Elevator). The strangest sign is labeled "Fernald", which is the real name of the hook-handed man, revealed in The Grim Grotto.
The Baudelaires are Jewish. There are hints in the books and Daniel Handler (Lemony Snicket) confirmed that they are Jewish. Thus, in the show, there is Jewish music at the wedding, to which Olaf dances a hora, and there are several Hebrew/Yiddish words in various episodes.
When Lemony is talking in the family library you can see focused points of light igniting the fires that eventually destroy the home. An allusion to the eventual cause of the fires, Count Olaf's magnifying device.