• Answer
• Race
  • Dwarf
  • Gnomes
• What the Players Handbook has to say
  • Fighter
  • Paladin
  • Ranger
  • Mage, Specialist Wizard e.g. Illusionist
  • Cleric
  • Druid
  • Thief
  • Bard

Answer

After initially trying to divine what classes can use which magic items by looking in the Players Handbook, I found that the final word on this is actually in the Dungeon Masters Guide.

The DMG, in summary, says :

• Armor
  • All classes except Wizards (because Wizards can’t wear any type of armor)
• Weapons
  • All classes
• Rings
  • All rings (except for the Ring of Wizardry) are usable by all classes (see DMG Table 91): “Magical rings can be worn and used by all character classes and humans/humanoids not specifically prohibited elsewhere.”
• Potions
  • Almost all potions are usable by all classes (3 are constrained to Warriors) (see DMG Table 89)
• Protection Scrolls
  • All classes “A protection scroll can be read by any class of character even without a read magic spell.”
• Scrolls
  • Wizards and Priests can use scrolls. Rogues can use them at 10th level and above : “Upon reaching 10th level, a bard can attempt to use magical devices of written nature– scrolls, books, etc. However, his understanding of magic is imperfect (although better than that of a thief), so there is a 15% chance that any written item he uses is read incorrectly.” and “At 10th level, a thief gains a limited ability to use magical and priest scrolls. A thief’s understanding of magical writings is far from complete, however. The thief has a 25% chance to read the scroll incorrectly and reverse the spell’s effect.”
• Rods
  • Each class can use different rods (see DMG Table 92)
• Staves
  • Most are usable by Priests or Wizards and some by all classes (see DMG Table 93)
• Wands
  • Wizards can use all wands. Some wands are usable by all classes (see DMG Table 94)
• Miscellaneous Magic
  • Each class can use different miscellaneous magic items (see DMG Table 95 - 104) : “Unless a description specifically restricts item use, or a letter representing a particular class follows a listing, items are usable by any class. Class letters are (C) clerics, (F) fighters, etc., and each listing includes appropriate sub-classes.”

This information is based on the Magic Item Tables section which says “Items followed by a group name are usable only by characters of that group.” which implies that items with no group indicated is usable by all classes.

Race

As a side not and for completeness, Dwarves and Gnomes have a chance of failure when using magical items :

Dwarf

“dwarves have trouble using magical items. All magical items that are not specifically suited to the character’s class have a 20% chance to malfunction when used by a dwarf… except weapons, shields, armor, gauntlets, and girdles”

Gnomes

“Gnomes also suffer a 20% chance for failure every time they use any magical item except weapons, armor, shields, illusionist items, and (if the character is a thief) items that duplicate thieving abilities”

What the Players Handbook has to say

Fighter

“The disadvantage warriors have is that they are restricted in their selection of magical items and spells.” (Warrior)
“they can use many magical items, including potions, protection scrolls, most rings, and all forms of enchanted armor, weapons, and shields.”

Paladin

“The disadvantage warriors have is that they are restricted in their selection of magical items and spells.” (Warrior)
“may not possess more than 10 magical items. Furthermore, these may not exceed one suit of armor, one shield, four weapons (arrows and bolts are not counted), and four other magical items”

Ranger

“The disadvantage warriors have is that they are restricted in their selection of magical items and spells.” (Warrior)
“nor is he ever able to use priest scrolls or magical items unless specially noted otherwise.”

Mage, Specialist Wizard e.g. Illusionist

“Wizards can use more magical items than any other characters. These include potions, rings, wands, rods, scrolls, and most miscellaneous magical items. A wizard can use a magical version of any weapon allowed to his class but cannot use magical armor, because no armor is allowed.” (Wizard)

Cleric

“They can use a fair number of magical items including priest scrolls, most potions and rings, some wands and rods, staves, armor, shields, and magical versions of any weapons allowed by their order.”

Druid

“Unlike the cleric, the druid is allowed to use only “natural” armors - padded, hide, or leather armor and wooden shields, including those with magical enhancements”
“Druids can use all magical items normally allowed priests, except for those that are written (books and scrolls) and armor and weapons not normally allowed for druids.”

Thief

“They are allowed to use a wide variety of magical items, weapons, and armor” (Rogue)
“Use Scrolls: At 10th level, a thief gains a limited ability to use magical and priest scrolls. A thief’s understanding of magical writings is far from complete, however. The thief has a 25% chance to read the scroll incorrectly and reverse the spell’s effect.”

Bard

“They are allowed to use a wide variety of magical items, weapons, and armor” (Rogue)
“Upon reaching 10th level, a bard can attempt to use magical devices of written nature– scrolls, books, etc. However, his understanding of magic is imperfect (although better than that of a thief), so there is a 15% chance that any written item he uses is read incorrectly.”