No uniform winner
Squared-error loss is \ell(\theta,t)=(\theta-t)^2. Show that if the parameter space contains at least two points, no estimator has risk R(\theta,T) smaller than every other estimator at every \theta.
Suppose \theta_0 and \theta_1 are two different points of the parameter space, and suppose an estimator T^\star minimised the squared-error risk at every \theta. The constant estimator T_0\equiv\theta_0 has risk 0 at \theta_0 and risk (\theta_1-\theta_0)^2 at \theta_1. So R(\theta_0,T^\star)\le 0, which forces T^\star=\theta_0 with probability one when the true value is \theta_0. The constant T_1\equiv\theta_1 has risk 0 at \theta_1. If T^\star is also best at \theta_1, its risk there is 0, so T^\star=\theta_1 with probability one when the true value is \theta_1. One estimator cannot be equal to both constants. Hence no uniform winner exists as soon as the parameter space has two points.