Let \((M,g,\eta,\varphi,\xi)\) be a \((2n+1)\)-dimensional almost contact metric manifold, where \(g\) is a Riemannian metric, \(\eta\) is a smooth 1-form, \(\xi\) is the Reeb vector field and \(\varphi\) is \((1, 1)\)-tensor field. If there are smooth functions \((\alpha,\beta)\) satisfying \((\nabla \varphi)(X,Y) =\alpha\, (g(X,Y)\xi - \eta(Y)X) +\beta\, (g(\varphi X,Y)\xi - \eta(Y) \varphi X)\), then it is called a trans-Sasakian manifold of type \((\alpha,\beta)\). \textit{J. C. Marrero} has shown [Ann. Mat. Pura Appl. (4) 162, 77--86 (1992; Zbl 0772.53036)] that a trans-Sasakian manifold of dimension \(\geq 5\) is either cosymplectic (type \((0,0)\)), or \(\alpha\)-Sasakian (type \((\alpha,0)\)), or \(\beta\)-Kenmotsu (type \((0, \beta)\)). In this paper, the authors obtain necessary and sufficient conditions for a 3-dimensional compact and connected trans-Sasakian manifold of type \((\alpha,\beta)\) to be homothetic to a Sasakian manifold. They also show that if a compact trans-Sasakian manifold admits an isometric immersion in the Euclidean space \(\mathbb{R}^4\) with Reeb vector field \(\xi\) being a transformation of the unit normal vector field under the complex structure of \(\mathbb{R}^4\), then it is homothetic to a Sasakian manifold. The authors introduce the axiom of a torus for a 3-dimensional trans-Sasakian manifold: for each \(p\in M\), there exists an isometric immersion \(f:\mathbb{T}^2 \to M\) tangential to \(\xi\) and \(p\in f(\mathbb{T}^2)\). Then the authors show that a 3-dimensional connected trans-Sasakian manifold with Ricci curvature in the direction of the Reeb vector field being a nonzero constant, satisfying the axiom of a torus is homothetic to a Sasakian manifold.