Inspired by a conversation about interview coding tasks from a list I'm on, I present the following - I considered it too long to email there. It took me longer than I expected to write; my x86 assembly is quite rusty. I'm not claiming it's pretty, but it fits in a single sector and most of the overhead is actually ELF structures. 

; nasm -f elf fizzbuzz.asm
; ld -melf_i386 -s -o fizzbuzz fizzbuzz.o
; ./fizzbuzz

section .data

fizz	db	" fizz"
fizzlen	equ	$ - fizz
buzz	db	" buzz"
buzzlen	equ	$ - buzz
num	db	"   "
numend	equ	$ - 1
numlen	equ	$ - num
nl	db	0xa
nllen	equ	$ - nl

curnum	db 1

section .text

	global _start

_start:
	mov ax, [curnum]
	call printnum

	mov ax, [curnum]
	mov cx, 3
	xor dx, dx
	div cx
	cmp dx, 0
	jnz notfizz

	mov edx, fizzlen
	mov ecx, fizz
	call printstr

notfizz:
	mov ax, [curnum]
	mov cx, 5
	xor dx, dx
	div cx
	cmp dx, 0
	jnz notbuzz

	mov edx, buzzlen
	mov ecx, buzz
	call printstr

notbuzz:
	mov edx, nllen
	mov ecx, nl
	call printstr

	inc BYTE [curnum]
	cmp BYTE [curnum], 100
	jle _start

	xor ebx, ebx
	mov eax, 1
	int 0x80

printnum:
	mov edi, numend
	mov cx, 10
p1:
	xor edx, edx
	div cx
	add dx, '0'
	mov [edi], dl
	dec edi
	cmp ax, 0
	jne p1

	mov ecx, num
	mov edx, numlen
printstr:
	mov ebx, 1
	mov eax, 4
	int 0x80
	ret