Non-monotonic decrease of residuals in Conjugate Gradients:
The residual $\|Ax-b\|$ is not necessarily monotonically decreasing.
Check out Stephen Boyd's slides on the conjugate gradient method. He shows an example where it increases (slide 22). http://see.stanford.edu/materials/lsocoee364b/11-conj_grad_slides.pdf
However, the distance from the solution (in terms of the inner product defined by $A$) $$\|x-x^{\star}\|_{A}^{2} = (x-x^{\star})^{T} A (x-x^{\star})$$ is monotonically decreasing. This implies that the objective $f(x) = x^{T} A x - 2 b^{T} x$ is also monotonically decreasing since $$\|x-x^{\star}\|_{A}^{2} = x^{T} A x - 2 b^{T} x + \|x^{\star}\|_{A}^{2}$$ where $x^{\star}$ satisfies $A x^{\star} = b$.
Note the difference between the objective above and the squared residual $$\|A x - b\|^2 = x^T A^T A x - 2 b^T A x + b^T b$$
I suppose you want to solve $Ax=b$ with $A$ being a symmetric, positive definite matrix.
Convergence of CG is monotonic -- with respect to the norm that is induced by the $A$-weighted scalar product. That is, if $r_k:=Ax_k-b$ is the current residual, then $\|x_k-x\|_A = \|r_k\|_{A^{-1}}^2 := r_k^*A^{-1}r_k$ is a decreasing sequence.
At the same time $\|r_k\|^2=r_k^*r_k$ need not be monotonically decreasing.
As CG is a stable method, the influence of rounding errors is only recognizable when the norm residuals is close to machine precision.
However, I don't know about the CGS methods you mentioned...
I made an EDIT to my answer to correct a wrong statement. Actually, in the $A$ norm, only the errors decrease monotonically but not the residuals. Thanks to @fieres who pointed this out in his answer.
There's a mistake in the accepted answer by Jan.
It says $r_k^T Ar_k$ is monotonically decreasing. Really it should say $r_k^T A^{-1}r_k$ is monotonically decreasing. This quantity, however, cannot be tracked in practice since $A^{-1}$ is unknown. To check whether the algorithm works correctly, you should check the objective function $F(x_k)=1/2 x_k^T A x_k - k_x^T b$, which does decrease monotonically, too.
See http://see.stanford.edu/materials/lsocoee364b/11-conj_grad_slides.pdf for details.
I think pterojacktyl's answer should be the accepted one. It's correct and comprehensive.