AFM, Nanoindentation, MIcrotensile tester etc are few of the common equipment which can be used to study the mechanical properties of fibers 

Direct measurement is also a viable way. 

By using a suitable load cell and grip, you can measure the tensile strength and Young's modulus AF a fiber easily.

By Use of Tensometer equipment you can evaluate at least Tensile properties id fiber.

Or you may use single fiber pull out test also.

The longitudinal modulus, tensile strength, and ultimate tensile strain of single filaments or impregnated tows, as well as strands, can obtained experimentally as described in ASTM standarts D3379-89 and   D4018.

Either through testins fibers alone (certain tests dedicated to fiber performance evaluatin) or through testing fiber reinforced composites in fiber direction

caracterisation of fibres need firstly a tensile test

you must know jauge length there are  stress vs a longh effect

other tests are necessary  for example compatibilty with matrix

I concur with Dr. Neto's comments.  ASTM D4018 is the most appropriate evaluation method for continuous tow reinforcing fibers.  The alternative single filament test techniques are interesting when developing new fibers but are not as relevant when considering reinforcing effects.

Hidalgo, O. 2003. Bamboo the gift of the gods. In the figure 6.6 of this one book presents a form to test fibers to tension

use indentation test for bonding characteristics or push out test

Besides the direct standart  tests mentioned above by colleagues, I try to propose more advanced indirect method of identification of fibers mechanical characteristics using the experimental data obtained by testing of composite samples. For this goal you need  suitable model of fiber-reinforced composites which described in literature.     

the different theoretical and experimental models in the work of

- Aveston, COOPER AND KELLY

- BEAUMONT AND WELLS

- Thouless

With the use of Multi continuum theory, you can obtain the properties of constituents (fiber, matrix)  from composite properties. 

There are many techniques to evaluate fiber mechanical properties. But what is important is evaluation of mechanical properties of fibers / filaments as a part of composite. The best way is to evaluate mechanical properties of composites using experimental techniques, and then evaluate mechanical properties of fibers using inverse micro-mechanics.