Examples of processes that are reversible isentropic but not adiabatic? Since $ds=\frac{dq_{rev}}{T}$ for reversible processes it seems we can have reversible isentropic processes that are not adiabatic provided the temperature changes in such way that the sum of $\frac{dq}{T}$ is zero but the sum of $dq$ itself is not zero. Is this possible? What are some example of such a process used/found in real life?
 A: In conventional thermodynamic language "adiabatic" process means being adiabatic (no heat exchnage) at every instant of the process not just in the sense that the total heat exchange is zero. The process itself can be reversible or irreversible, only the heat exchanged must be zero. The so-called adiabatic legs of the Carnot cycle (isothermal-adiabatic-isothermal-adiabatic) are such processes during which there is no heat exchange at all; heat is exchanged during the two isothermal legs: absorb heat at the high temperature and release heat at the low temperature.
Using conventional language then adiabatic and reversible must mean that $\delta q_{rev}=0$, hence, you also have $dS=0$, see below the answer of @BobD; in other words the system entropy is kept constant at any instant, i.e, isentropic.
Except for a single book, namely Pippard: Elements of Classical Thermodynamics, every other book on thermodynamics under the sun uses the word adiabatic in the sense I described above. (In his book Pippard uses the word adiathermal in the sense that during any instant of the process that can be reversible or irreversible the heat exchanged is zero, and by adiabatic he means the special case of a process that is reversible and adiathermal that is isentropic.)
A: Any reversible cyclic process will be isentropic (since entropy is a state function), whereas it's not necessary that it's adiabatic as well. A famous example is the Carnot cycle.
A: 
it seems we can have reversible isentropic processes that are not
adiabatic..

No you cannot.
An isentropic process is by definition a process that is both adiabatic and reversible. So you can't have an isentropic process that is not adiabatic.
However, you can have an adiabatic process that is not isentropic, if it is not a reversible process. An example is an adiabatic process involving friction losses.
The equation defining a differential change in entropy
$$ds=\frac{dq_{rev}}{T}$$
can be used to determine the difference in entropy between any two states by assuming any convenient path between the two states and evaluating the integral, because entropy is a state function (independent of the path).
For example, suppose the actual process connecting the two states is an irreversible adiabatic process. Since it is irreversible, $ds$ is not zero and it is not isentropic even though there is no heat transfer.
We can determine the difference in entropy between the two states by evaluating the equation for any convenient reversible path between the two states since entropy is a state function. We might, for example, connect the two states with a combination of a reversible isothermal and reversible isochoric (constant volume) process, and evaluate the integral for both.
Hope this helps.
