Low Serial Numbers
Serial numbers that start with four or more zeros are fancy: meaning 00000001 - 00009999. The more leading zeros, the better.
High Serial Numbers
Serial numbers that start with four or more nines are fancy: meaning 99990000 - 99999999. The higher, the better. All notes are printed starting with serial number 1, but recently notes have NOT been printed to 99999999. Notes printed on sheets of 32 and 50 within the past few decades (uncut sheets excluded) are only printed to 96000000. Prior to that, block letters have rolled over at various increments - 99200000, 99840000, and 99999999.
For a recent note that has a block rollover at 96000000, you might think "that means 95999173 is rare because it's within 827 of the highest number, similar to five leading zeros." Maybe a collector might find it interesting to pay over face value for, but I think it would be considerably lower than a five leading zero number.
Each digit is one number higher or lower than the previous digit. Examples: 01234567, 98765432. Some people consider ladders fancy if they go up and down like: 23456765, or wrap around from zero: 56789012.
Binary serial numbers contain only two unique digits (order doesn't matter). Example: 74774774
True binary serial numbers are a kind of binary that contain only zeros and ones - like machine code. Example: 01001101
Trinary serial numbers are like binaries, except that they contain only three unique digits. Again, order doesn't matter. Example: 79557977
The first half and second half of the serial number are the same. Examples: 74627462
A super repeater is when the first two digits are repeated four times. Example: 39393939.
In a quad double, the first four digits are the same number, and the last four digits are the same number. Examples: 55553333, 11116666
Radar serial numbers are the same forward as it is backwards. It is named this because the word "r-a-d-a-r" is spelled the same forward and back. Examples: 12533521, 49366394
There are also more types of radars:
The radar is also a repeater, meaning the first four digits and last four digits are the same as well as the entire number being the same forwards and backwards. These types of radars are inherently also binary. Examples: 34433443, 61166116
The radar only consists of two digits. Examples: 83833838
Every digit in the number is the same. Examples: 11111111, 88888888
Seven of the same digit are in a row. Example: 77777771
Similar to 7-in-a-row except that the lone digit is somewhere in the middle of the number. Example: 77177777
Serial numbers can contain dates. The format might change depending on your locale, but a number like 07041776 or 02221732 could indicate the finalization of the Declaration of Independence, or George Washington's birthday. Some people try to collect numbers matching their birthdays or other life milestones.