What is the strongest way to fasten the shelf to the wall?

Fasten your shelf supports directly to wall studs whenever possible for the most secure attachment. Use a stud finder to locate studs in a wall.

thespruce.com - Properly Space Your Shelves and Wall Supports - The Spruce

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There are two important considerations when installing wall shelves: first, making sure the material used for the shelves has an appropriate span capacity; and second, considering the vertical spacing between the shelves—a decision that is based on the types of objects you will store on the shelves. To prevent the shelves from sagging under the weight, you need to know the relative span limitations of different shelving materials. "Span" in this instance means the distance between supports for the shelf. Span limits are based on the strength of the material—the stronger the material, the more space you can put between wall supports. You may still be able to store fairly heavy objects on shelving materials with a short span limit, but you will need to support the shelf with support brackets that are spaced closer together. Vertical shelf spacing is simply a matter of deciding what you will store on the shelves, and space the shelves accordingly. Tip Fasten your shelf supports directly to wall studs whenever possible for the most secure attachment. Use a stud finder to locate studs in a wall. When it is impossible or impractical to attach shelf supports to studs, you must make sure to use the proper wall anchors.

Span Limits

The span limit is defined as the maximum distance apart that you can place the shelving supports. Glass, particleboard, solid lumber, plywood, and other common shelving materials can have span limits that can range greatly. The load that the shelf carries will, of course, affect that allowable span, but for the following span recommendations, it is assumed that the shelf will support a full load of standard-sized hardback books.

Plywood Shelves

Plywood for shelves should be 3/4 inch thick or thicker. Standard sheets of plywood come in 1/4-inch, 3/8-inch, 1/2-inch, 5/8-inch, and 3/4-inch thicknesses. Plywood panels over 3/4 inch in thickness are usually special order products. Thinner sheets of plywood can be glued together (laminated) to make a thicker panel for cutting sturdier shelves. The longest recommended span between wall supports for 3/4-inch-thick plywood shelves is 36 inches.

1 x Lumber Shelves

1 x lumber is standard building grade boards, such as nominal 1x8, 1x10, or 1x12 boards. Be aware that the actual thickness of standard lumber is less than the nominal measurement by which it is categorized. All 1 x lumber is about 3/4 inch thick. When choosing 1 x lumber for shelves, it's best to use select-grade lumber, which has fewer knots and other flaws that can weaken the wood. The longest recommended span between wall supports for 1x10 lumber is 24 inches.

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The longest recommended span between wall supports for 1x12 lumber is 28 inches.

2 x Lumber Shelves

2 x dimension lumber, such as 2x10s or 2x12s, actually measures about 1 1/2 inches thick. As with 1 x lumber, it's best to choose select-grade material that has fewer knots and will make stronger, more attractive shelves than construction-grade material.

The longest recommended span between wall supports for 2x10 lumber is

48 inches.

48 inches. The longest recommended span between wall supports for 2x12 lumber is

56 inches.

Particleboard Shelves

Particleboard is a manufactured product that is made from sawdust and other wood by-products bonded together with glues and resins. It is commonly sold in thicknesses of 3/8 inch, 1/2 inch, 5/8 inch, and 3/4 inch. Like plywood, particleboard can be glued and screwed together to create thicker shelves. Single-layer shelving made from particleboard should be at least 5/8 inch thick. The longest recommended span between wall supports for 5/8-inch particleboard is 24 inches. The longest recommended span between wall supports for 3/4-inch particleboard is 28 inches.

Glass Shelves

Spans for glass shelves are based on the type, thickness, and overall size of the glass. Common types of glass for shelves are annealed and tempered. Tempered glass is much stronger than annealed, and it breaks into small pieces when shattered, while annealed glass breaks into large shards. Given all the factors involved, it's best to calculate a shelf's weight rating rather than relying on a basic span dimension when installing glass shelves. A glass supplier can recommend a safe weight for the glass you choose. Two examples are provided here: The maximum weight rating for 1/4-inch-thick x 11-inch-wide annealed glass supported every 24 inches is 29 pounds. The maximum weight rating for 1/4-inch-thick x 11-inch-wide tempered glass supported every 24 inches is 117 pounds.

Recommended Shelf Height Spacing

Use the following guidelines to determine the "clear shelf height" needed for the types of items you plan to place on your shelves. Clear shelf height is the space measured from the top of the lower shelf to the underside of the shelf above it.