Number of semistable subbundles of a semistable bundle Is the following true ? If so, is there a quick proof of it ? (Perhaps using the uniqueness of the graded object associated to a Jordan-Holder filtration or maybe otherwise)
Suppose $E$ is an $\omega$-semistable bundle with slope $\mu$ over a compact Kahler manifold $(X,\omega)$. There are only finitely many (upto isomorphism) $\omega$-semistable subbundles of $E$ having slope equal to $\mu$.
 A: That is not true.  The issue has to do with nontrivial extensions between semistable bundles of the same slope.  If you have a compact Kähler manifold where every semistable sheaf of slope $0$ has vanishing $H^1$, then I suspect that it is true that there are only finitely many isomorphism classes occuring for semistable subbundles of the same slope.  However, for manifolds where semistable sheaves of slope $0$ can have large $H^1,$ there are counterexamples.  The proposition below gives one class of counterexamples.
Let $k$ be a field, e.g., the field of complex numbers.  Let $X$ be a smooth, projective, geometrically connected $k$-curve of genus $g.$
Lemma. There exists a short exact sequence of $\mathcal{O}_X$-modules,$$e:\ \ 0\to \mathcal{O}_X \xrightarrow{q} E \xrightarrow{p}  H^1(X,\mathcal{O}_X)\otimes_k \mathcal{O}_X \to 0,$$ such that the connecting map of the long exact sequence, $$ \partial_e:H^1(X,\mathcal{O}_X)\otimes_k H^0(X,\mathcal{O}_X)\to H^1(X,\mathcal{O}_X),$$ is the identity map.  The locally free $\mathcal{O}_X$-module $E$ is semistable of slope $0$ and rank $g+1.$
Proof.  Choose a $k$-basis, $$e_1,\dots,e_g\in \text{Ext}^1_k(\mathcal{O}_X,\mathcal{O}_X) = H^1(X,\mathcal{O}_X).$$ Each element $e_i$ gives a Yoneda extension class of a short exact sequence, $$e_i: \ \ 0 \to \mathcal{O}_X \xrightarrow{q_i} E_i \xrightarrow{p_i} \mathcal{O}_X \to 0.$$  Each element $E_i$ is semistable (not polystable) of slope $0$ and of rank $2.$  Moreover, the connecting map $\delta_{e_i}$ has image equal to the span of $e_i$ in $H^1(X,\mathcal{O}_X).$  
The direct sum $F= E_1\oplus \dots \oplus E_g$ is semistable of slope $0$ and rank $2g.$  There is a short exact sequence, $$\oplus e_i:\ \ 0 \to \bigoplus_{i=1}^g \mathcal{O}_X \xrightarrow{\oplus q_i}  \bigoplus_{i=1}^g E_i \xrightarrow{\oplus p_i} \bigoplus_{i=1}^g \mathcal{O}_X \to 0. $$  The connecting map of this short exact sequence is the direct sum of the connecting maps $\delta_{e_i}.$  For the first term in the short exact sequence, there is also a "summing" surjective $\mathcal{O}_X$-module homomorphism, $$\Sigma:\bigoplus_{i=1}^g \mathcal{O}_X \to \mathcal{O}_X.$$  Denote the kernel by $K=\text{Ker}(\Sigma).$  Define $E$ to be the quotient of $F$ by $K.$  Thus, there is a short exact sequence, $$e:\ \ 0 \to \mathcal{O}_X \xrightarrow{q}  E \xrightarrow{\oplus p_i} \bigoplus_{i=1}^g \mathcal{O}_X \to 0. $$  Again, since the first and third terms are semistable of slope $0$, also the middle term $E$ is semistable of slope $,$ and of rank $g+1.$  Finally, the connecting map $\delta_e$ is the composition of the direct sum of the summands $\partial_{e_i}$ and the summing map.  Thus, the connecting map is an isomorphism.  QED
Now assume that $g\geq 1.$ For every nonzero element $a\in H^1(X,\mathcal{O}_X),$ consider the subbundle $E_a$ of $E$ that is the pullback by the following map, $$a:\mathcal{O}_X\to H^1(X,\mathcal{O}_X)\otimes_k\mathcal{O}_X.$$  In other words, there is a commutative diagram of short exact sequences, $$\begin{array}{cccccccccc} e_a: & 0 & \to & \mathcal{O}_X & \xrightarrow{q} & E_a & \to & \mathcal{O}_X & \to & 0 \\ & & & \text{Id}~\downarrow & & \downarrow{f_a} & & \downarrow{a} \\ e: & 0 & \to & \mathcal{O}_X & \xrightarrow{q} & E & \xrightarrow{\oplus p_i} & H^1(X,\mathcal{O}_X)\otimes_k\mathcal{O}_X & \to & 0  \end{array}.$$  By the Snake Lemma, $f_a$ is injective with locally free cokernel, i.e., $f_a(E_a)$ is a subbundle of $E$ isomorphic to $E_a$, a semistable bundle of slope $0,$ and of rank $2.$  Note that the image of $f_a$ equals the image of $f_{\lambda\cdot a}$ for every $\lambda\in k^\times.$  Thus, the subsheaf of $E$ depends only on the $k$-point $[a]=[\text{Span}(a)]$ in the projective $k$-space $\mathbb{P}H^1(X,\mathcal{O}_X) \cong \mathbb{P}^{g-1}_k$ parameterizing rank $1$ subspaces of $H^1(X,\mathcal{O}_X).$
Proposition. Every semistable subbundle of $E$ of slope $0$ and rank $2$ equals $E_a$ for a unique $k$-point $[a]\in \mathbb{P}H^1(X,\mathcal{O}_X).$  For $k$-points $[a]$ and $[b],$ the $\mathcal{O}_X$-module $E_a$ is isomorphic to $E_b$ if and only if $[a]$ equals $[b].$
Proof.  For every semistable, rank $2$ subsheaf $G$ of slope $0$, consider the short exact sequence, $$0\to (G\cap q(\mathcal{O}_X)) \to G \to p(G) \to 0.$$  As a subsheaf of the locally free sheaf $H^1(X,\mathcal{O}_X)\otimes_k \mathcal{O}_X$, the sheaf $p(G)$ is torsion-free, hence locally free (since $X$ is a smooth curve).  As a quotient of a semistable sheaf of slope $0$, also $p(G)$ has slope $\geq 0$ and rank equal to either $1$ or $2$, depending on whether the rank of the subsheaf $G\cap q(\mathcal{O}_X)$ of $q(\mathcal{O}_X)$ equals $1$ or $0.$  Since $H^1(X,\mathcal{O}_X)\otimes_k \mathcal{O}_X$ is semistable of slope $0$, also $p(G)$ has slope $\leq 0.$  Thus, $p(G)$ has slope $0.$  It follows that $p(G)$ is a direct summand of $H^1(X,\mathcal{O}_X)\otimes_k \mathcal{O}_X.$  Since this sheaf is polystable, $p(G)$ is of the form $V\otimes_k \mathcal{O}_X$ for a $k$-subspace $V$ of $H^1(X,\mathcal{O}_X)$ of rank $1$ or $2$.  Finally, if the rank of $V$ is $2$, then $G\cap q(\mathcal{O}_X)$ is zero, so that $G\to p(G)$ is an isomorphism.  But then $V=H^0(X,G)$ is a $k$-subspace of $H^0(X,E)$ of rank $2$, whereas $k=H^0(X,q(\mathcal{O}_X))\to H^0(X,E)$ is an isomorphism of $k$-vector spaces by the construction of $E$.  This is a contradiction.  Therefore $p(G)$ has rank $1$.  This forces the subsheaf $G\cap q(\mathcal{O}_X)$ of $q(\mathcal{O}_X)$ to be of rank $1$ and of slope $0$, thus equal to the entire sheaf $q(\mathcal{O}_X).$  Thus $V$ equals $\text{Span}(a)$ for some nonzero $a$, and $G$ equals $E_a.$
Associated to the short exact sequence $e_a$, consider the long exact sequence of cohomology, $$0 \to H^0(X,\mathcal{O}_X) \to H^0(X,E_a) \to H^0(X,\mathcal{O}_X) \xrightarrow{\partial_a} H^1(X,\mathcal{O}_X).$$  Since $a$ is nonzero, the first map above is an isomorphism, and the last map above is an injection.  Thus, the subsheaf $q(\mathcal{O}_X)$ is the intrinsic subsheaf $H^0(X,E_a)\otimes_k \mathcal{O}_X,$ the cokernel by this intrinsic sheaf is isomorphic to $\mathcal{O}_X,$ and the connecting map $\delta$ is identified with the intrinsic connecting map, $$\partial_{E_a}:H^0(X,E_a/H^0(X,E_a)\otimes_k\mathcal{O}_X) \to H^1(X,H^0(X,E_a)\otimes_k \mathcal{O}_X), \text{ i.e.,}$$
$$\partial_{E_a}:H^0(X,E_a/H^0(X,E_a)\otimes_k\mathcal{O}_X) \to H^0(X,E_a)\otimes_k H^1(X,\mathcal{O}_X).$$  Of course the image of $\partial_a$ in $H^1(X,\mathcal{O}_X)\setminus \{0\}$ gives a well-defined and intrinsic element in the projective space, $$\mathbb{P} H^1(X,\mathcal{O}_X) \cong \mathbb{P}^{g-1}_k.$$  Thus, the sheaf $E_a$ is isomorphic to $E_b$ if and only if the elements $[a],[b]\in \mathbb{P} H^1(X,\mathcal{O}_X)$ are equal.  QED
Finally, if $k$ is infinite and if $g\geq 2,$ then there are infinitely many $k$-points of $\mathbb{P}^{g-1}_k.$  Thus, there are infinitely many isomorphism classes of rank-$2$, slope-$0$ semistable subsheaves $f_a(E_a)$ of the semistable, rank-$(g+1)$, slope-$0$ sheaf $E$.
