(Original Guides by Tweaker, Puto, and RetroKoH)
Source: SCHG Page
Original Commit: 7d4da8d
Updated Commit: 561325a
Ground Speed Cap can be toggled in S1Fixed by setting GroundSpeedCapEnabled to 0 or 1.
Air Speed Cap can be toggled in S1Fixed by setting AirSpeedCapEnabled to 0 or 1.
Rolling Speed Cap can be toggled in S1Fixed by setting RollSpeedCapEnabled to 0 or 1.

Here's a breakdown of the speed cap from the Sonic Physics Guide, courtesy of Lapper:

In Sonic 1, while Sonic is already running at a higher speed than he can possibly achieve on his own (such as having been bounced by a Spring on a wall), if you press in the direction he's moving, v_sonspeedacc will be added to obInertia as usual. However, notice that obInertia now exceeds v_sonspeedmax, which means obInertia will be set to v_sonspeedmax. Thus it is possible to curtail your forward momentum by pressing in the very direction of your motion. This can be solved in your engine (and was fixed in Sonic 2 and beyond) by checking to see if obInertia is less than v_sonspeedmax before adding v_sonspeedacc. Only if obInertia is already less than top will it check if obInertia exceeds v_sonspeedmax.

NOTE: I edited this explanation with variable names from the disassembly.

Ground Speed Cap

(Original guide by Tweaker)
Right, so let's remove that speed cap. Open _incObj/Sonic Move.asm and go to the label loc_1309A. Add these three lines:

loc_1309A:
	sub.w	d5,d0
	move.w	d6,d1
	neg.w	d1
	cmp.w	d1,d0
	bgt.s	loc_130A6
+	add.w	d5,d0
+	cmp.w	d1,d0
+	ble.s	loc_130A6
	move.w	d1,d0

This takes care of the speed cap when moving on the ground to the left, now to handle ground movement to the right. Find the label loc_13104. Add these three lines:

loc_13104:
	add.w	d5,d0
	cmp.w	d6,d0
	blt.s	loc_1310C
+	sub.w	d5,d0
+	cmp.w	d6,d0
+	bge.s	loc_1310C
	move.w	d6,d0

Air Speed Cap

(Original guide by Puto)
We're not done just yet. We have one more to remove. This time we're opening _incObj/Sonic JumpDirection.asm. Just before loc_13278, immediately after the branch to that very label, we are going to add three lines that should look familiar:

	bgt.s	loc_13278
+	add.w	d5,d0
+	cmp.w	d1,d0
+	ble.s	loc_13278
	move.w	d1,d0

loc_13278:

This removes the airborne speed cap to the left, now to remove the other. We are already at label loc_13278, so let's just add these three lines:

loc_13278:
	btst	#bitR,(v_jpadhold2).w	; is right being pressed?
	beq.s	Obj01_JumpMove		; if not, branch
	bclr	#0,obStatus(a0)
	add.w	d5,d0
	cmp.w	d6,d0
	blt.s	Obj01_JumpMove
+	sub.w	d5,d0
+	cmp.w	d6,d0
+	bge.s	Obj01_JumpMove
	move.w	d6,d0

Obj01_JumpMove:

That handles that! Air Speed Cap removed.

Rolling Speed Cap

(Original guide by RetroKoH)
Devon recently gave us a bugfix that gave us a consistent rolling speed cap. Yes, that's right! There is a third speed cap! I've since added a toggle to S1Fixed that allows us to disable this speed cap as well. To do this, apply the rolling cap fix, then remove the highlighted lines:

loc_131CC:
	move.b	obAngle(a0),d0
	jsr	(CalcSine).w
	move.w  obInertia(a0),d2
	muls.w  d2,d0
	asr.l   #8,d0
-	cmpi.w  #$1000,d0
-	ble.s   .checkNegY
-	move.w  #$1000,d0

-.checkNegY:
-	cmpi.w  #-$1000,d0
-	bge.s   .setVelocityY
-	move.w  #-$1000,d0

.setVelocityY:
	move.w  d0,obVelY(a0)		
	muls.w  d2,d1
	asr.l   #8,d1
-	cmpi.w  #$1000,d1
-	ble.s   .checkNegX
-	move.w  #$1000,d1

-.checkNegX:
-	cmpi.w  #-$1000,d1
-	bge.s   .setVelocityX
-	move.w  #-$1000,d1

.setVelocityX:
	move.w  d1,obVelX(a0)
	bra.w	loc_1300C
; End of function Sonic_RollSpeed