All linear exponential smoothing state space models have at least one unit root. ETS(A,N,N) and ETS(A,Ad,N) have one unit root, ETS(A,A,N) has two unit roots, ETS(A,N,A) and ETS(A,Ad,A) both have one unit root and a seasonal unit root, while ETS(A,A,A) has two unit roots and a seasonal unit root.

For the correspondence between the linear ETS models and ARIMA models, see http://otexts.org/fpp2/arima-ets.html

See chapter 11 of my Springer book (www.exponentialsmoothing.net) for the detailed derivations.

The multiplicative models do not have an equivalent ARIMA form, so it is not really possible to talk about them having unit roots in the same way.

All linear exponential smoothing state space models have at least one unit root. ETS(A,N,N) and ETS(A,Ad,N) have one unit root, ETS(A,A,N) has two unit roots, ETS(A,N,A) and ETS(A,Ad,A) both have a seasonal unit root, while ETS(A,A,A) has one unit root and a seasonal unit root.

For the correspondence between the linear ETS models and ARIMA models, see http://otexts.org/fpp2/arima-ets.html

See chapter 11 of my Springer book (www.exponentialsmoothing.net) for the detailed derivations.

The multiplicative models do not have an equivalent ARIMA form, so it is not really possible to talk about them having unit roots in the same way.