When reading the books or papers on p-adic Hodge theory, non trivial example of p-adic representation seems to be only the example of Tate curves. To be sure, I had read the very readable introduction by L.Berger and there are some calculations for elliptic curves by ($\varphi$, Fil)-modules. But, are there more concrete examples (coming from given equations) which one can calculate by hand(s)?

Not sure what you mean by "coming from given equations". If you're looking for concrete examples, you can start with abelian representations of $G_K = Gal(K^{alg}/K)$ where $K$ is a finite extension of $Q_p$. Choose a Lubin-Tate character $\chi_K : G_K \to O_K^\times$ and see what kinds of representations you can get by looking at $\eta \circ \chi_K$ where $\eta : O_K^\times \to O_L^\times$ is a character, with $L/K$ a finite extension, and you map $O_L^\times \to GL_d(Z_p)$ using a basis of $O_L$. What Hodge-Tate-Sen weights can you get this way? Then what extensions can there be between such representations? Even the abelian representations can teach you quite a bit.
Any $p$-adic modular form has a $p$-adic representation and any classical modular form has p-adic reprsentation such that it is potentially semi-stable at $p$.
Generally, any proper smooth scheme $X$ over $\mathbb{Q}$ yields a $p$-adic representation given by the action of $G_\mathbb{Q}$ on the $p$-adic etale cohomology $H^{i}(X\times \overline{\mathbb{Q}}, \mathbb{Z}_p) \otimes \mathbb{Q}_p$, and it is always potentially semi-stable at $p$.
• Example of what ?? there is a lot of examples of $(\varphi,\Gamma)$-module but what do you mean by $(\varphi, Fil)$? what is $Fil$? – Adel BETINA Feb 1 '16 at 8:38