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Maybe the easiest way to find a good example is in complete intersections in weighted projective spaces.

Let us consider a 3-fold $X$ of the form of complete intersection $X=X_{d_1,\ldots, d_c}\subset\mathbb{P}(a_0, \ldots, a_n)$ with the following properties:

1. $n-c=3$, i.e. $\dim X=3$.

2. $\sum d_i=\sum a_j+1$, i.e., $\mathcal{O}_X(K_X)=\mathcal{O}_X(1)$.

3. $X$ has at most terminal singularities.

Note that such 3-fold $X$ satisfies that $h^1(\mathcal{O}_X)=h^2(\mathcal{O}_X)=0$ and $h^3(\mathcal{O}_X)=h^0(K_X)=h^0(\mathcal{O}_X(1))$. So in order to find examples you want, we just need to find such $3$-fold with $a_0>1$ (this is equivalent to $h^0(\mathcal{O}_X(1))=0$), and take a resolution.

There are many such $3$-folds, the list is given by Iano-Fletcher in Section 15 of [A. R. Iano-Fletcher, Working with weighted complete intersections, Explicit birational geometry of 3-folds. London Mathematical Society, Lecture Note Series, 281. Cambridge University Press, Cambridge, 2000. 101–173]. So we just check the list with to find $3$-fold with $a_0>1$.

Here I mention one of my favorite example: $X=X_{46}\subset\mathbb{P}(4,5,6,7,23)$. The reason why it is interesting is that for this $X$, $|mK_X|$ gives a birational map when $m\geq 27$, but $|26K_X|$ is not birational. This is so far the only known example of 3-fold of general type with this property (that $|26K_X|$ is not birational).

Maybe the easiest way to find a good example is in complete intersections in weighted projective spaces.

Let us consider a 3-fold $X$ of the form of complete intersection $X=X_{d_1,\ldots, d_c}\subset\mathbb{P}(a_0, \ldots, a_n)$ with the following properties:

1. $n-c=3$, i.e. $\dim X=3$.

2. $\sum d_i=\sum a_j+1$, i.e., $\mathcal{O}_X(K_X)=\mathcal{O}_X(1)$.

3. $X$ has at most terminal singularities.

Note that such 3-fold $X$ satisfies that $h^1(\mathcal{O}_X)=h^2(\mathcal{O}_X)=0$ and $h^3(\mathcal{O}_X)=h^0(K_X)=h^0(\mathcal{O}_X(1))$. So in order to find examples you want, we just need to find such $3$-fold with $a_0>1$ (this is equivalent to $h^0(\mathcal{O}_X(1))=0$), and take a resolution.

There are many such $3$-folds with properties 1--3, the list is given by Iano-Fletcher in Section 15 of [1], where he classified all possible cases for $a_d\leq 100$ and $d\leq 2$. One interesting thing is that Iano-Fletcher's list is later proved to be a full classification of $3$-folds with properties 1--3 by Chen--Chen--Chen [2]. So we just check the list with to find $3$-fold with $a_0>1$.

Here I mention one of my favorite example: $X=X_{46}\subset\mathbb{P}(4,5,6,7,23)$. The reason why it is interesting is that for this $X$, $|mK_X|$ gives a birational map when $m\geq 27$, but $|26K_X|$ is not birational. This is so far the only known example of 3-fold of general type with this property (that $|26K_X|$ is not birational).

References: [1] A. R. Iano-Fletcher, Working with weighted complete intersections, Explicit birational geometry of 3-folds. London Mathematical Society, Lecture Note Series, 281. Cambridge University Press, Cambridge, 2000. 101–-173

[2] J-J. Chen, J.A. Chen, M. Chen, On quasismooth weighted complete intersections, J. Algebraic Geom. 20 (2011): 239--262.

Maybe the easiest way to find a good example is in complete intersections in weighted projective spaces.

Let us consider a 3-fold $X$ of the form of complete intersection $X=X_{d_1,\ldots, d_c}\subset\mathbb{P}(a_0, \ldots, a_n)$ with the following properties:

1. $n-c=3$, i.e. $\dim X=3$.

2. $\sum d_i=\sum a_j+1$, i.e., $\mathcal{O}_X(K_X)=\mathcal{O}_X(1)$.

3. $X$ has at most terminal singularities.

Note that such 3-fold $X$ satisfies that $h^1(\mathcal{O}_X)=h^2(\mathcal{O}_X)=0$ and $h^3(\mathcal{O}_X)=h^0(K_X)=h^0(\mathcal{O}_X(1))$. So in order to find examples you want, we just need to find such $3$-fold with $a_0>1$ (this is equivalent to $h^0(\mathcal{O}_X(1))=0$), and take a resolution.

There are many such $3$-folds, the list is given by Iano-Fletcher in Section 15 of [A. R. Iano-Fletcher, Working with weighted complete intersections, Explicit birational geometry of 3-folds. London Mathematical Society, Lecture Note Series, 281. Cambridge University Press, Cambridge, 2000. 101–173]. So we just check the list with to find $3$-fold with $a_0>1$.

Here I mention one of my favorite example: $X=X_{46}\subset\mathbb{P}(4,5,6,7,23)$.

Maybe the easiest way to find a good example is in complete intersections in weighted projective spaces.

Let us consider a 3-fold $X$ of the form of complete intersection $X=X_{d_1,\ldots, d_c}\subset\mathbb{P}(a_0, \ldots, a_n)$ with the following properties:

1. $n-c=3$, i.e. $\dim X=3$.

2. $\sum d_i=\sum a_j+1$, i.e., $\mathcal{O}_X(K_X)=\mathcal{O}_X(1)$.

3. $X$ has at most terminal singularities.

Note that such 3-fold $X$ satisfies that $h^1(\mathcal{O}_X)=h^2(\mathcal{O}_X)=0$ and $h^3(\mathcal{O}_X)=h^0(K_X)=h^0(\mathcal{O}_X(1))$. So in order to find examples you want, we just need to find such $3$-fold with $a_0>1$ (this is equivalent to $h^0(\mathcal{O}_X(1))=0$), and take a resolution.

There are many such $3$-folds, the list is given by Iano-Fletcher in Section 15 of [A. R. Iano-Fletcher, Working with weighted complete intersections, Explicit birational geometry of 3-folds. London Mathematical Society, Lecture Note Series, 281. Cambridge University Press, Cambridge, 2000. 101–173]. So we just check the list with to find $3$-fold with $a_0>1$.

Here I mention one of my favorite example: $X=X_{46}\subset\mathbb{P}(4,5,6,7,23)$. The reason why it is interesting is that for this $X$, $|mK_X|$ gives a birational map when $m\geq 27$, but $|26K_X|$ is not birational. This is so far the only known example of 3-fold of general type with this property (that $|26K_X|$ is not birational).

Maybe the easiest way to find a good example in complete intersections in weighted projective spaces.

Let us consider a 3-fold $X$ of the form of complete intersection $X=X_{d_1,\ldots, d_c}\subset\mathbb{P}(a_0, \ldots, a_n)$ with the following properties:

1. $n-c=3$, i.e. $\dim X=3$.

2. $\sum d_i=\sum a_j+1$, i.e., $\mathcal{O}_X(K_X)=\mathcal{O}_X(1)$.

3. $X$ has at most terminal singularities.

Note that such 3-fold $X$ satisfies that $h^1(\mathcal{O}_X)=h^2(\mathcal{O}_X)=0$ and $h^3(\mathcal{O}_X)=h^0(K_X)=h^0(\mathcal{O}_X(1))$. So in order to find examples you want, we just need to find such $3$-fold with $a_0>1$ (this is equivalent to $h^0(\mathcal{O}_X(1))=0$), and take a resolution.

There are many such $3$-folds, the list is given by Iano-Fletcher in Section 15 of [A. R. Iano-Fletcher, Working with weighted complete intersections, Explicit birational geometry of 3-folds. London Mathematical Society, Lecture Note Series, 281. Cambridge University Press, Cambridge, 2000. 101–173]. So we just check the list with to find $3$-fold with $a_0>1$.

Here I mention one of my favorite example: $X=X_{46}\subset\mathbb{P}(4,5,6,7,23)$.

Maybe the easiest way to find a good example is in complete intersections in weighted projective spaces.

Let us consider a 3-fold $X$ of the form of complete intersection $X=X_{d_1,\ldots, d_c}\subset\mathbb{P}(a_0, \ldots, a_n)$ with the following properties:

1. $n-c=3$, i.e. $\dim X=3$.

2. $\sum d_i=\sum a_j+1$, i.e., $\mathcal{O}_X(K_X)=\mathcal{O}_X(1)$.

3. $X$ has at most terminal singularities.

Note that such 3-fold $X$ satisfies that $h^1(\mathcal{O}_X)=h^2(\mathcal{O}_X)=0$ and $h^3(\mathcal{O}_X)=h^0(K_X)=h^0(\mathcal{O}_X(1))$. So in order to find examples you want, we just need to find such $3$-fold with $a_0>1$ (this is equivalent to $h^0(\mathcal{O}_X(1))=0$), and take a resolution.

There are many such $3$-folds, the list is given by Iano-Fletcher in Section 15 of [A. R. Iano-Fletcher, Working with weighted complete intersections, Explicit birational geometry of 3-folds. London Mathematical Society, Lecture Note Series, 281. Cambridge University Press, Cambridge, 2000. 101–173]. So we just check the list with to find $3$-fold with $a_0>1$.

Here I mention one of my favorite example: $X=X_{46}\subset\mathbb{P}(4,5,6,7,23)$.