You've asked a question that will get you some interesting answers, for pure effectiveness as an Infantry Weapon the M1859 Sharps in .54 (the .52 was a civilian model that did see some service but it was less common) was an outstanding weapon w/ few if any equals in accuracy or dependability. The Spencer was a robust & well liked rifle & carbine, though it's cartridge was notorious for poor stopping power. The Henry was certainly the most technologically advanced weapon of the time but it suffered w/ the same problem as the Spencer. At much past 100 yards neither the Spencer or Henry were very effective... Though the sheer volume of firepower they could generate was mind boggling for the time.
The most important thing to remember though is that the man made the rifle, not the other way around. While the Sharps was an excellent rifle, there just weren't that many of them. The Springfiled & Enfield were the killers of the time, w/ .58 minnie balls fired from them doing the lions share of killing.
THere is of coarse quite a bit of argument as to which was better: Springfield or Enfield... it's a tough question to answer as both had comparable accuracy, knock down power & reliability. It's generally argued that the sites on an Enfield are Superior to those on a Springfield but the Springfiled seemed to handle field use better and benefited from FAR superior quality control at the production line. Having fired repro's of both and an original Springfield... I'd say they handle the same w/ the differences being so minor as to make it all but impossible to decide on which is the superior model.
Aphillbilly mentions the Whitworth... while it was quite accurate there were so few of them that their contribution was quite minor. Another rifle of the same vein was the Kerr, actually a more accurate and longer ranged weapon than the Whitworth, and considerably rarer. A period Kerr might easily fetch $100,000 today. (Actually Aphillbilly there is quite a bit of argument still as to if the bullet that killed Sedgwick was a from a Whitworth. I've seen convincing arguments both ways and have to admit I don't know on that score. Regardless, it was one hell of a shot, better than 700 yards.)
Both were hideously expensive, a Whitworth costing the
CSA $1000 a piece while a P1858 or 53 Enfield was less than $15... There are reasonable arguments that the Whitworth was not really any more effective than the standard Enfield.
As to the age old argument between the 10 pdr Parrot & the 3" Ordinance Rifle.... The Parrot had a tendency to burst in front of the reinforcing band and the 3" Rifle was all together simpler to make and cheaper IIRC as well as having the same accuracy. Frankly, I think their performance was pretty much the same, though the 3" Rifle could push a bolt an extra fifty or so yards. The accuracy that those two cannon could accomplish is really quite frightening. There are stories of a Union Gunner w/ a 3" Rifle picking out individual flag bearers and hitting them... at better than 1300 yards. The casualty numbers from Cannon are very misleading, loaded w/ Canister a Napoleon was as terrifying a weapon as they came. At Fredricksburg Cannon did the lions share of the killing. It was a massive psych weapon... If you've ever been on a re-enactment field you'll understand what I mean... modern re-enactors actually are allowed to use only about a half charge of powder.
The LeMat was an excellent weapon (J.E.B. Stuart carried two on his horses pommel), but it's weight made it very unpopular. The various Colts were quite popular & had a steller reputation but those that carried them swore by the Remington New Model Army. Having live fired all of them I favor the Remington, I didn't care for the balance of the LeMat, though I could appreciate the added firepower.
THe bayonet was more of a psych weapon as well w/ only a few bayonet charges really being effective... though those succesful ones are quite impressive... 1st MN & 20th Maine at Gettysburg, John Kings Regulars at Chickamaugua. THough it's much easier to come up w/ terrifying slaughters because of bayonet charges. Frankly, the CW rifle makes a better club than a spear.
As to swords, very few were sharpened at all, though there are some notable exceptions. The
cav sabre was used more for a club to break the coller bone or the neck of a man as you rode past him. Shooting accurately from a moving horse is easier said than done, which is why it wasn't often done. By the end of the war
Cav was being used more as mounted Infantry than in traditional
Cav charges.
Cav's effectiveness as recon was too valuable to use them as Mounted Infantry... though again there were notable exceptions.
To add to Aphillbilly's point about disease, a man was three times more likely to succumb to disease than to an enemy bullet. Dysentery being one of the biggest killers.
That's my take on the issue... for an excellent book on the firearms of the time I strongly suggest Edwards, William B., "Civil War Guns," The Stackpole Company, 1962. It's in it's third or fourth printing now, runs about $45 but is well worth the investment... it's one of the few books I keep by my side religiously.