A lot has been said of the 3rd and 5th Confederate Infantry regiments, and some on Dearing's 8th Cavalry. However, what of the other Confederate regiments?
As far as I know, 1st Georgia Confederate served in the Army of Tennessee under Wilson, Clements, and H. R. Jackson, 2nd Confederate (25th Ms;aka The Valley Regiment) fought at Shiloh before disbanding, and was led by John C. Moore before his promotion.
As for other 'Confederate' units, haven't heard much.
I don't know much of the specific histories of many of the units with the designation of "Confederate" in their unit names, but here is a listing of I believe all the known units of that type in the Confederate armies:
CONFEDERATE ARMY UNITS
NOTES: Although a Confederate Regular Army was authorized, only a few companies were actually raised. Most units carrying the "Confederate" designation in their unit names did so simply because they were comprised of companies from more than one Confederate state. Note that all formal Confederate Engineer units carry the "Confederate" unit name designation.
CONFEDERATE ARTILLERY UNITS
1st Confederate Artillery Battery (see 1st Louisiana Regular Artillery Battery)
1st Confederate (Fitzgerald's) Artillery Regiment (did not complete organization)
White's (Confederate) Horse Artillery Battery (see White's (Tennessee) Artillery Battery)
CONFEDERATE CAVALRY UNITS
1st Confederate Cavalry Regiment
Maj. Henry C. Bate
Col. John T. Cox
Col. H. Clay King
Lt. Col. C.S. Robertson
1st Confederate Cavalry Regiment Regulars (Co. A)
Capt. Edward Ingraham
Col. Edmund Kirby Smith
Col. Earl Van Dorn
1st Confederate Trans-Mississippi Cavalry Battalion
Maj. Thompson J. Bird
3rd Confederate Cavalry Regiment
Lt. Col. F.M. Corn
Col. W.N. Estes
Col. James R. Howard
Lt. Col. John McCaskill
Col. P.H. Rice
6th Confederate Cavalry Regiment
Capt. Thomas Claiborne
Maj. H. Clay King
Capt. John F. Lay
Lt. Col. James A. Pell
Maj. M.J. Wicks
6th Confederate Cavalry Battalion
Lt. Col. George M. Jessee
Maj. A.L. McAfee
7th Confederate Cavalry Regiment
Lt. Col. Thomas D. Claiborne
Col. William C. Claiborne
Maj. Jesse H. Sikes
Col. V.H. Taliaferro
7th Confederate Cavalry Battalion
Lt. Col. Clarence J. Prentice
Maj. W.G. Repass
8th Confederate (Wade's) Cavalry Regiment
Lt. Col. Jefferson Falkner
Col. Joseph H. Field
Lt. Col. John S. Prather
Col. William B. Wade
Maj. John T. Wright
10th Confederate Cavalry Regiment
Col. Charles T. Goode
Maj. John B. Rudolph
Lt. Col. Miles M. Slaughter
Col. William I. Vason
11th Confederate Cavalry Regiment (see 3rd Cavalry Regiment)
12th Confederate Cavalry Regiment (see 6th Cavalry Regiment)
13th Confederate Cavalry Regiment (see 3rd Cavalry Regiment)
14th Confederate Cavalry Regiment
Lt. Col. John B. Cage
Col. F. Dumonteil
Lt. Col. William H. Garland
Maj. Pinkney C. Harrington
15th Confederate Cavalry Regiment
Col. Henry Maury
Lt. Col. T.J. Myers
Maj. Robert H. Partridge
16th Confederate Cavalry Regiment (see 6th Cavalry Regiment)
20th Confederate Cavalry Regiment (see Lay's Mississippi Cavalry Regiment)
Baxter's (Confederate) Cavalry Battalion (see Baxter's (Mississippi) Battalion)
Gillum's (Confederate) Regiment, Mounted Riflemen (did not complete organization)
Dearing's (Confederate) Cavalry Regiment
Col. James Dearing
Harman's (Confederate) Cavalry Regiment (also see Harman's (Mississippi) Cavalry Regiment)
Col. B. Desha Harman
Mead's (Confederate) Cavalry Regiment
Col. Lemuel H. Mead
CONFEDERATE ENGINEERS
1st Confederate Engineers Regiment
Lt. Col. William W. Blackford
Maj. Peyton Randolph
Col. Thomas M.R. Talcott
1st Confederate Engineers Battalion
Lt. Col. Hugh T. Douglas
Maj. R.P. Rowley
2nd Confederate Engineers Regiment
Maj. D. Wintter
3rd Confederate Engineers Regiment
Maj. John W. Green
Lt. Col. S.W. Presstman
4th Confederate Engineers Regiment
Col. Hugh T. Douglas
CONFEDERATE INFANTRY UNITS
1st Confederate Infantry Regiment (also see 1st Georgia (Confederate) Infantry Regiment)
Lt. Col. Jacob W. Aderhold
Maj. Elijah M. Dodson
Lt. Col. James C. Gordon
Maj. William S. Lovell
Col. George A. Smith
Col. John B. Villepigue
1st Confederate Foreign Legion Infantry Regiment (see Tucker's Infantry Regiment)
1st Confederate Infantry Battalion
Lt. Col. George H. Forney
Lt. Col. Francis B. McClung
Maj. Lawrence W. O'Bannon
1st Confederate Foreign Infantry Battalion
Lt. Col. Julius G. Tucker
2nd Confederate Infantry Regiment (also see 25th Mississippi Infantry Regiment)
Maj. Thomas H. Mangum
Col. John D. Martin
Lt. Col. Edward F. McGehee
2nd Confederate Infantry Battalion
Maj. James C., Jr. Malone
2nd Confederate Foreign Infantry Battalion (see 8th Infantry Battalion)
2nd Confederate Foreign Legion Infantry Battalion (see 8th Infantry Battalion)
3rd Confederate Infantry Regiment
Lt. Col. John F. Cameron
Lt. Col. J.C. Cole
Lt. Col. James B. Johnson
Lt. Col. Henry V. Keep
Col. John S. Marmaduke
4th Confederate Infantry Regiment (also see 1st AL, MS, and TN Infantry Regiment)
Col. Alpheus Baker
Maj. Joseph Barbiere
Lt. Col. John A. Minter
Maj. Thaddeus H. Shackelford
4th Confederate Infantry Battalion (see Confederate Guards Response Battalion (Louisiana))
5th Confederate (Smith's) Infantry Regiment
Lt. Col. James C. Cole
Maj. Richard J. Person
Col. James A. Smith
5th Confederate (Walker's) Infantry Regiment (see 40th Tennessee Infantry)
5th Company, CS Retributors Infantry
Lt. Bennett H. Young
8th Confederate Infantry Battalion
Lt. Col. Garnett Andrews
Maj. R.T. Fouche
9th Confederate Infantry Regiment (see 5th (Smith's) Infantry Regiment)
Bradford's (Confederate) Infantry Battalion
Maj. J.D. Bradford
Brook's (Confederate) Infantry Battalion
Maj. John H. Brooks
Brush (Confederate) Infantry Battalion
Unknown
Tucker's (Confederate) Infantry Regiment
Col. Julius G. Tucker