Well I guess the most famous is the problem of suffering.
If God is loving why doesn't he save people from natural disasters and illness? Humans would do this, in the name of compassion if they could; God doesn't.
If God is love, why doesn't he stop rape and murder? Don't say free will. Police stopping a criminal isn't a violation of free will, it's a restriction of liberty.
If you say that God's love is significantly different from what we mean by love, why call God loving?
I'm no philosopher, but this argument easily defeats itself.
If we assume God is all-knowing, and the creator of everything, the it stands to reason He both created love and fully understands it.
In this sense, He would be like a professor 100% versed in quantum physics, trying to teach a kindergarten class. As the student, His concepts seem so irrational, and require so much more foundational knowledge, that we assume He's just speaking nonsense. When, in reality, we have such a ways to go in our education that we can't even begin to understand.
Sin exists because people wish to be free from God. This means, whether you love God or not, bad things are bound to happen. Just like when your child wants to do something on their own, and you can see it ending badly, but you stay out of it so they learn a lesson. Most would agree that a crucial part of parenting is to not shelter your children. You understand that experiencing the real world brings wisdom.
The Lord loves those whom He corrects. As we are His children, He both loves and corrects us. He allows us to go out on our own, no matter how painful the consequences. These are usually the experiences that bring us to our knees.
Now, as far as natural disasters, war, and plagues caused by God in the Old Testament, these were all disciplinary actions taken after years of warnings. Noah warned people for 100 years before the flood, but nobody wanted to believe in God. He sent nine different plagues that also proved His power as God before the final plague of taking the firstborn, and that was only the firstborn of those who deliberately chose to not show their belief by refusing to put blood on the door. Then, God even let his own people be taken into slavery and lost in the desert because they still fell away from Him. But, at the same time, He brought them out of tribulation as soon as they turned back to Him.
The wars He ordered in the OT were to wipe out horrifically evil nations. These nations would sacrifice their children, gouge out the eyes of their captives, slowly impale criminals on large poles, and practice many other tortures and sinful actions. God gave multiple warnings to repent, but they didn't. He told the Jews He wasn't giving the Promised land over to them because they were so good, it was because the inhabiting people were so bad that they had to be wiped out.
A loving God created a perfect world. Then, an evil devil convinced people it wasn't good enough. We create evil by desiring to live apart from God.