Arduino Uno with ESP8266-01 with AT
If you want to use esp-01 over SoftwareSerial, you must set the AT firmware's default baud rate to 9600 baud. The default baud rate of AT firmware is 115200 baud. SoftwareSerial can't work at this speed.
To change the baud rate you must have a way to communicate with AT firmware at the default baud rate of 115200 baud. You can use the USB adapter of the Uno. Upload BareMinimal sketch from Examples to Uno to make it silent and connect the esp-01 to RX and TX pins of the Uno.
Connect RX to RX and TX to TX to have the cross connection to USB of the Uno. Then connect from Serial Monitor to esp-01 at 115200 with &quot;Both line end characters&quot; set. Try some basic AT commands and then still at 115200 baud send the command to change the default baud rate AT+UART_DEF=9600,8,1,0,0

In setup() in sketch begin for SoftwareSerial is missing. Add  esp8266.begin(9600);

Uno pin 11 is RX of software serial, pin 10 is TX. The RX pin 11 must connect to TX pin of esp-01. Same applies for TX pin 10. It must connect to RX pin of the esp-01.
On the Fritzing schema on Uno pin 11 is green wire and on esp-01 TX is green wire. But the green wire is on sides of the level converter not on the same position. The conclusion is that you have RX connected to RX not to TX. Switch wires on best at level converter.

The 3.3 V pin of some Uno clones can't supply enough current for WiFi operations of the esp8266. You need to convert to 3.3 V from Uno 5 V pin or use a separate 3.3 V power source for the esp-01.